EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1878-1893 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Reflexive Condition on Skew Monoid Rings Eltiyeb Ali1,2 1 Department of Mathematics, Faculty of Education, University of Khartoum, Sudan 2 Department of Mathematics, Collage of Science and Arts, Najran University, Saudi Arabia Abstract. This paper is devoted to introducing and studying two concepts, σ-skew strongly M - reflexive and σ-skew strongly M -nil-reflexive on monoid rings, which are generalizations of strongly M -reflexive and M -compatible. The paper covers the basic properties of skew monoid rings of the form R∗M . It is shown that if R is a left APP (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M -reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M -nil-reflexive. Additionally, under some necessary and sufficient conditions, a skew monoid ring R ∗M is proven to be σ-skew strongly M -nil-reflexive when σ :M → Aut(R) is a monoid homomorphism. Furthermore, if R is a left APP , then the upper triangular matrix ring Tn(R) is σ̄-skew strongly M -nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject. 2020 Mathematics Subject Classifications: 16S34, 16S36, 20M25 Key Words and Phrases: Left APP -ring, skew monoid ring R ∗M , quasi Armendariz ring, σ-skew strongly M -nil-reflexive ring 1. Introduction Throughout this article, R and M denote an associative ring with identity and a monoid, respectively. Mason introduced the reflexive property for ideals and this con- cept was generalized by some authors, defining idempotent reflexive right ideals and rings, completely reflexive, weakly reflexive (see namely, [13], [15] and [24]). Let R be a ring and I be a right ideal of R. In [24], I is called a reflexive right ideal if for any x, y ∈ R, xRy ⊆ I implies yRx ⊆ I. The reflexive right ideal concept is also specialized to the zero ideal of a ring, namely, a ring R is called reflexive [24], if its zero ideal is reflexive and a ring R is called completely reflexive if for any x, y ∈ R, xy = 0 implies yx = 0. Reduced rings are completely reflexive and every completely reflexive ring is semicommutative. The notion of Armendariz ring is introduced by Rege and Chhawchharia [21]. They defined a ring R to be Armendariz if f(x)g(x) = 0 implies aibj = 0 for all polynomials f(x) = a0 + a1x+ a2x 2 + · · ·+ amx m and g(x) = b0+ b1x+ b2x 2+ · · ·+ bnxn ∈ R[x]. In [21] a ring R is called semicommutative if for all x, y ∈ R, xy = 0 implies xRy = 0. This is equivalent to the definition that any DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4827 Email addresses: eltiyeb76@gmail.com, emali@nu.edu.sa (E. Ali) https://www.ejpam.com 1878 © 2023 EJPAM All rights reserved. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1879 left (right) annihilator of R is an ideal of R. An ideal I of a ring is called semiprime if xRx ⊆ I implies x ∈ I for x ∈ R and R is called semiprime if 0 is a semiprime ideal. It should be noted that every semiprime ideal is reflexive, as can be easily verified and therefore every ideal of a fully idempotent ring (i.e., a ring where I2 = I for all ideals I) is reflexive according to [6]. The ring R is said to be weakly reflexive if xry = 0 implies yrx is nilpotent for x, y ∈ R and all r ∈ R. The rings without nonzero nilpotent elements are said to be reduced rings. According to [9] a ring R is called quasi-Armendariz if whenever polynomials f(x) = a0 + a1x + a2x 2 + · · ·+ amx m, g(x) = b0 + b1x+ b2x 2 + · · ·+ bnx n ∈ R[x] satisfy f(x)R[x]g(x) = 0, then aiRbj = 0 for each i, j. It was proved in [10], if R is an Armendariz ring, then R is completely reflexive if and only if R[x] is completely reflexive. According to [22], a ring R is σ-skew nil M -McCoy if αβ ∈ Nil(R) ∗ M there exist a nonzero element c ∈ R such that aiσgi(c) ∈ Ni(R) for each i, where α = a1g1 + a2g2 + · · · + amgm and β = b1h1 + b2h2 + · · · + bnhn are nonzero element in R ∗M . In [3] the author defined “nil skew generalized power series reflexive rings" for which f, g ∈ [[RS,≤, ω]] satisfying fhg ∈ [[nil(R)S,≤, ω]] implies that ghf ∈ [[nil(R)S,≤, ω]]. Similarly, in [1] the author discussed “ the nilpotent elements and nil-reflexive property of generalized power series rings", where fhg ∈ [[nil(R)S,≤]] implies that ghf ∈ [[nil(R)S,≤]] for f, g, h ∈ [[RS,≤]]. The investigation of the composition of the collection of nilpotent elements in noncommutative ring constructions is a crucial and highly active field in noncommutative algebra. This is evidenced by numerous studies conducted by various authors see [3], [1], [22], [15], [13], [4], [5], [2] and [18]. This paper is devoted to examining the nilpotent elements found in skew monoid rings. In this context, let R be a ring and M be a u.p.-monoid. It is assumed that M oper- ates on R through a homomorphism that maps to the automorphism group of R. This homomorphism is denoted as σ : M → Aut(R), i.e., a module homomorphism from M to the group of automorphisms of R, here, an automorphism of R is a ring isomorphism from R to itself, so, Aut(R) is the group of all such isomorphisms. For any given element g ∈ M , the notation σg denotes the image of g under the module homomorphism σ, i.e., σg = σ(g) ∈ Aut(R). In other words, σg is an automorphism of R that depends on the choice of the element g ∈ M . By using the monoid homomorphism σ, we can create a skew monoid ring denoted as R∗M . This ring consists of finite formal combinations of ele- ments in M , represented as Σg∈Mxgg. Multiplication in this ring is induced by the formula (xgg)(yhh) = (xgσg(yh))(gh). Therefore, R ∗M is a ring that is free as a left R-module with basis M . A commonly accepted fact is that if a polynomial f(x) is defined over a commutative ring, then it is nilpotent if only if the coefficient of f(x) is too. However, it should be noted that this statement does not hold true for noncommutative rings. The researcher introduced and studied two concepts, σ-skew strongly M -reflexive and σ-skew strongly M -nil-reflexive on monoid rings. The paper covers the basic properties of skew monoid rings of the form R∗M . It is shown that if R is a left APP (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M -reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M -nil-reflexive. Furthermore, under certain conditions, a skew monoid ring R ∗M is proven to be σ-skew strongly M - E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1880 nil-reflexive when σ :M → Aut(R) is a monoid homomorphism. Additionally, it is proved that a ring R is σ-skew strongly M -nil-reflexive if and only if R/I is σ̄-skew strongly M -nil-reflexive. Consequently, if R is a left APP , then the upper triangular matrix ring Tn(R) is σ̄-skew strongly M -nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject. To recall, a monoid M is referred to as a unique product (u.p.)-monoid if, for any two non-empty finite subsets X and Y of M , there exist an element x ∈ X and an element y ∈ Y such that their product xy is distinct from the product of any other pair (u, v) ∈ X × Y , i.e., (u, v) ̸= (x, y) implies uv ̸= xy. The element xy is termed as a u.p.-element of the set XY = {pq : p ∈ X, q ∈ Y }. Unique product monoids and groups have significant implications in ring theory, particularly in providing a positive solution to the zero-divisor problem for group rings. Their structural properties have been extensively studied in literature (see references [17, 19]). The category of monoids that fall under the classification of u.p.-monoids is both extensive and significant. This category encompasses monoids that are either right or left totally ordered, submonoids of a free group, and torsion-free nilpotent groups. For a positive integer n, let Matn(R) denote the ring of all n×n matrices and Tn(R) the ring of all n×n upper triangular matrices with entries in R. We write R[x] and Sn(R), for the polynomial ring over a ring R and the subring consisting of all upper triangular matrices over a ring R with equal main diagonal entries. 2. Reflexive-Type Properties in Skew Monoid Rings In this section, we discuss various constructions and extensions under which the class of σ-skew strongly M -reflexive rings is closed. By Definition 2.3 [18], a monoid homomor- phism σ :M → Aut(R) is called compatible if the ring R is σg-compatible for each g ∈M , that is, xy = 0 ⇔ xσg(y) = 0 for all x, y ∈ R. Now we have the following generalization of reflexive. Definition 1. We say that a ring R is σ-skew strongly M -reflexive (σ-skew strongly re- flexive relative to a monoid M), if φ(R ∗M)ψ = 0 implies that biσgi(Rσs(aj)) = 0, where φ = b1g1 + b2g2 + · · ·+ bngn and ψ = a1h1 + a2h2 + · · ·+ amhm are nonzero elements in R ∗M, then ψ(R ∗M)φ = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ m. Definition 2. In [20]. A ring R is called strongly M -reflexive, if whenever φ,ψ ∈ R[M ] with φR[M ]ψ = 0, then ψR[M ]φ = 0. In [16], Nasr-Isfahani and Moussavi introduced a ring R with an endomorphism σ and defined it as σ-weakly rigid if the condition xRy = 0 holds if and only if xσ(Ry) = 0 for any x, y ∈ R. A ring R is σ-rigid if and only if R is σ-compatible and reduced by [8]. According to [16], any prime ring that has an automorphism σ is considered to be σ-weakly rigid. If a monoid homomorphism σ :M → Aut(R) is weakly-rigid (compatible), it means that the ring R is also weakly rigid (compatible) with respect to each g ∈ M under the automorphism σg. The following example illustrates that the compatibility of σ is necessary. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1881 Example 1. Let S be any nonzero reversible ring and M be a monoid generated by an element ρ such that ρ has infinite order. Suppose R = S ⊕ S with the usual addition and multiplication, and define σ : M → Aut(R) such that σρ((x, y)) = (y, x). Then the ring R is reflexive and M is u.p.-monoid, but σ is not compatible, since (1, 0)(0, 1) = (0, 0) whereas (1, 0)σρ((0, 1)) = (1, 0). Now we will prove that the ring R is not σ-skew strongly M -reflexive. For, let φ = (1, 0)e + (1, 0)g and ψ = (0, 1)e − (1, 0)g be nonzero elements in R ∗M and any ϕ ∈ R ∗M. Then we can easily see that φϕψ = 0, but ψϕσg(φ) ̸= 0. Therefore, R is not σ-skew strongly M -reflexive. Theorem 1. Let R be a ring, M be a monoid generated by an element ρ such that ρ has infinite order and σ :M → Aut(R) a compatible monoid homomorphism given by σρ = ψ. Suppose N be any monoid with an element of infinite order. If the skew monoid ring R∗M is a strongly N -reflexive, then R is σ-skew strongly M -reflexive. Proof. Let φ = ∑m i=1 aigi ψ = ∑n 1 bjgj be nonzero elements in R ∗ M such that φϕψ = 0 for any ϕ = ∑v r=1 ℓrgr ∈ R ∗M. Then, for each 1 ≤ k ≤ m + v + n, we have ck = ∑ i+r+j=k aiσgi(ℓrσs(bj)) = 0 for s ∈ M. Now, let h ∈ N such that O(h) = ∞ and define F,G ∈ (R ∗M)[N ] as in the following: F = (a1eM )eN + (a1g)h + (a2g2)h2 + · · · + (amgm)hm and G = (b1eM )eN + (b1g)h + (b2g2)h2 + · · · + (bngn)hn. Since φ, ϕ and ψ are nonzero in R ∗ M, so F and G are nonzero elements in (R ∗ M)[N ]. Moreover, from φϕψ = 0 and compatibility of σρ = ψ, one can easily obtain that FHG = 0 for any H ∈ (R∗M)[N ]. Since the skew monoid ring R∗M is stronglyN -reflexive. Then, aictbj = 0 for each t and so aiσgi(Rσs(bj)) = 0 for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, s ∈ M. By a compatible automorphism, we have bjσgj (Rσs(ai)) = 0. Therefore, R is σ-skew strongly M -reflexive. An ideal I of R is said to be right s-unital if, for each a ∈ I there exist an element e ∈ I such that ae = a. Note that if I and J are right s-unital ideals, then so is I ∩ J (if a ∈ I ∩ J, then a ∈ aIJ ⊆ a(I ∩ J)). We say a ring R is a left APP -ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. The following result follows from Tominaga Theorem 1 [23]. Lemma 1. An ideal I of a ring R is left (resp. right) s-unital if and only if for any finitely many elements a1, a2, . . . , an ∈ I, there exists an element e ∈ I such that ai = eai(resp. ai = aie) for each i = 1, 2, . . . , n. Lemma 2. (Lemma 1.13 [14]). Let M and N be u.p.-monoids. Then so is the monoid M ×N. Lemma 3. (Example 2.2 [7]). Let R be a ring and M be a monoid with an element of a finite order n ≥ 2. Let φ = Σn−1 i=0 g i and ψ = e− g, where |g| = n. Then φψ = 0 and so R is not M -nil-Armendariz. Lemma 4. (Lemma 1.1 [5]). Assume M is a u.p.-monoid. Then M is cancellative (i.e., for g, h, x ∈M, if gx = hx or xg = xh, then g = h). E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1882 Proposition 1. Let R be a ring, M be a u.p.-monoid and σ : M → Aut(R) a compatible monoid homomorphism. If R is a reduced left APP -ring, then R is σ-skew strongly M - reflexive. Proof. Suppose φ,ψ ∈ R∗M such that φ(R∗M)ψ = 0 implies that biσgi(Rσs(aj)) = 0 for any s ∈M, where φ = b1g1+b2g2+· · ·+bngn and ψ = a1h1+a2h2+· · ·+amhm ∈ R∗M. We shall prove that ψ(R ∗M)φ = 0, (i.e., ajσhj (Rσs(bi)) = 0 for all i, j. Let r be an arbitrary element of R. Then we have the following equation: (b1g1 + b2g2 + · · ·+ bngn)(rs)(a1h1 + a2h2 + · · ·+ amhm) = 0. (2.1) We proceed by induction on both n and m for every s ∈M and for 1 ≤ i ≤ n, 1 ≤ j ≤ m. If m = 1, then ψ = a1h1. Thus 0 = (b1g1+ b2g2+ · · ·+ bngn)(rs)(a1h1) = b1σg1(σs(ra1))+ b2σg2(σs(ra1)) + · · · + bnσgn(σs(ra1)) for every r ∈ R. By Lemma 4, M is a cancellative monoid. Thus gish1 ̸= gjsh1 for gi ̸= gj . Then biσgi(σs(ra1)) = 0. Hence bi ∈ ℓR(σs(Ra1)). By hypothesis, R is left APP , ℓR(Ran) is left s-unital by Lemma 1. Hence, there exist em ∈ ℓR(Ran) such that anem = an since σgi is an automorphism, i = 1, 2, . . . , n. Now suppose that m ≥ 2. Since M is u.p.-monoid, there exists p, q with 1 ≤ p ≤ n and 1 ≤ q ≤ m such that gpshq is uniquely presented by considering two subsets {g1, g2, . . . , gn} and {sh1, sh2, . . . , shm} of M. Thus, from φ(rs)ψ = 0 it follows that bpσgp(σs(raq)) = 0 and so bpσgp(σs(raq)) = 0. Thus σgp(cpσs(raq)) = 0 for c ∈ R, which implies that cpσs(raq) = 0 for every r ∈ R since σgp is an automorphism. Hence, cp ∈ ℓR(σs(Raq)). Since ℓR(σs(Raq)) is pure as a left ideal of R by Lemma 1, there exist an element eq ∈ ℓR(σs(Raq)) such that cp = cpeq. Thus, for every r ∈ R, we have 0 = φ(eqrs)ψ = (b1g1 + b2g2 + · · ·+ bngn)(eqrs)(a1h1 + a2h2 + · · ·+ amhm) = (b1g1 + b2g2 + · · ·+ bngn)(eqrs)(a1h1 + a2h2 + · · ·+ aq−1hq−1 + aq+1hq+1 + · · ·+ amhm) = (b1g1 + b2g2 + · · ·+ bngn)((eqrσg(aq))ghq) = (b1σg1(eq)g1 + b2σg2(eq)g2 + · · ·+ bnσgn(eq)gn)(rs) · (a1h1 + a2h2 + · · ·+ aq−1hq−1 + aq+1hq+1 + · · ·+ amhm). (2.2) Moreover, since biσgi(eq) = σgi(cieq) by induction, it follows that cieq ∈ ℓR(Rσs(aj)) for i = 1, 2, . . . , n, j = 1, 2, . . . , q−1, q+1, . . . ,m. Therefore, cp = cpeq ∈ ∩mj=1 ℓR(Rσs(aj)). Now we have bpσsp(Rσs(aj)) = σsp(bpRσs(aj)) = 0. For every gi ∈ M, since σgi is an au- tomorphism of R and σgq(R) = R, we obtain bpσgp(Rσgsc(aj)) = 0 for any j = 1, 2, . . . ,m. Thus, from φ(rs)ψ = 0 it follows that 0 = (b1g1 + b2g2 + . . . + bp−1gp−1 + bp+1gp+1 + · · · + bngn)(rs)(a1h1 + a2h2 + · · · + amhm). By using the previous method, there exist k ∈ {1, 2, . . . , p− 1, p+1, . . . , n} such that ck ∈ ∩mj=1ℓR(σs(Raj)). Thus, bkσgk(σs(Raj)) = σgk(ckσs(Raj)) = 0 for any j = 1, 2, . . . ,m. Hence (b1g1+b2g2+ · · ·+bp−1gp−1+bp+1gp+1+ · · · + bk−1gk−1 + bk+1gk+1 + · · · + bngn)(rs)(a1h1 + a2h2 + · · · + amhm) = 0. Continu- ing this procedure yields c1, c2, . . . , cn ∈ ∩mj=1ℓR(σs(Raj)) for every s ∈ M. Thus, bk ∈ ℓR(σs(Raj)) for any k = 1, 2, . . . , n, j = 1, 2, . . . ,m. Using induction on m + n, we obtain biσgi(σs(Raj)) = 0. Thus, ajσhj (σs(Rbi)) = 0 since R is reduced. Therefore, R is σ-skew strongly M -reflexive. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1883 Let (M,≤) be an ordered monoid. If for any g1, g2, h ∈ M, g1 < g2 implies that g1h < g2h and hg1 < hg2, then ≤ is called a strictly ordered monoid. Corollary 1. Let R be an M -ring, where M is a strictly totally ordered monoid and σ :M → Aut(R) is a monoid homomorphism. If R is quasi Armendariz, then R is σ-skew strongly M -reflexive. Proof. Let ϕ = Σni=1bigi and ψ = Σmj=1ajhj ∈ R ∗M satisfying ϕ(R ∗M)ψ = 0 implies that biσgi(Rσs(aj)) = 0 for any s ∈M and any i, j. We write (b1g1 + b2g2 + · · ·+ bngn)(rs)(a1h1 + a2h2 + · · ·+ amhm) = 0. (2.3) With g1 < g2 < . . . < gn, h1 < h2 < . . . < hm.We will use transfinite induction on a strictly totally ordered set ≤ to show that ψ(R ∗M)ϕ = 0. If we take m = 1 in Eq. (2.3), then we have (b1g1 + b2g2 + · · ·+ bngn)(rs)(a1h1) = 0. Therefore, we obtain biσgi(Rσs(a1)) = 0 for each 1 ≤ i ≤ i. Since M is a strictly totally ordered monoid, we have g1h1 < gih1 ≤ gihj = g1h1 for i ̸= 1 or j ̸= 1. Thus, Eq. (2.3) becomes (b1g1 + b2g2 + · · ·+ bngn)(rs)(a2h2 + a3h3 + · · ·+ amhm) = 0. (2.4) The case n = 1 is proved by similar argument. By applying the induction hypothesis for all 2 ≤ i ≤ n and 2 ≤ j ≤ m. Suppose that biσgi(Rσs(aj)) = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ m with λ ∈M is such that for any gi and hj , gihj < λ. We will show that biσgi(Rσs(aj)) = 0 for any gi and hj with gihj = λ. Set X = {(gi, hj)|gihj = λ}. Then X is a finite set. We write X as {(giq , hjq)|q = 1, 2, . . . , d} such that gi1 < gi2 < . . . < gid . Since M is cancellative, gi1 = gi2 and gi1hj1 = gi2hj2 = λ imply hj1 = hj2 since ≤ is a strict order, gi1 < gi2 and gi1hj1 = gi2hj2 = λ imply hj2 < hj1 . Thus we have hjd < hjd−1 < . . . < hj2 < hj1 . Now ∑ (gi,hj)∈X biσgi(rσs(aj)) = ∑d q=1 biqσgiq (rσs(ajq)) = 0. For any q ≥ 2, gi1hjq < giqhjq = λ, and so bi1σgiq (rσs(ajq)) = 0 by induction hypothesis. Thus, bi1σgiq (rajq) = 0 because R is M -rigidness and σ is automorphism. For the case where n ≥ 2 for M is cancellative. We can repeat this process to show that biσgi(Rσs(aj)) = 0 for all s ∈ M and all i, j. Consequently, we can see that ajσhj (Rσs(bi)) = 0 for all s ∈ M , 1 ≤ j ≤ m, 1 ≤ i ≤ n by rigidness and σ is auto- morphism. Thus, ψ(R ∗M)ϕ = 0. Therefore, R is σ-skew strongly M -reflexive. Proposition 2. Let R be a ring, M be a strictly totally ordered monoid and σ : M → Aut(R) a compatible monoid homomorphism. If R is semiprime, then R is σ-skew strongly M -reflexive. Proof. Since a semiprime ring is quasi-Armendariz and so reflexive, the proof follows from Corollary 1. For a ring R and n ≥ 2, let Vn(R) be the ring of all n×n upper triangular matrices over R that are constant on the diagonal. Let σ : M → Aut(R) be a monoid homomorphism. For each g ∈M,σ can be extended to a monoid homomorphism σ from M to Aut(Vn(R)) defined by σ((aij)) = (σg(aij)). E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1884 Theorem 2. Let R be an M -rigid ring, where M is a monoid and σ : M → Aut(R) is a monoid homomorphism, let n ≥ 2. If R is a left APP -ring, then Vn(R) is σ-skew strongly M -reflexive. Proof. Suppose that R is a left APP -ring and let φ = A1g1 + A2g2 + · · · + Angn and ψ = B1h1 + B2h2 + · · · + Bmhm ∈ Vn(R) ∗M such that φ(Vn(R) ∗M)ψ = 0. We use (a1, a2, . . . , an) ∈ Vn(R), where Ai =  a1 (i) a2 (i) a3 (i) · · · an (i) 0 a1 (i) a2 (i) · · · an−1 (i) 0 0 a1 (i) · · · an−2 (i) ... ... ... . . . ... 0 0 0 · · · a1 (i)  , Bj =  b1 (j) b2 (j) b3 (j) · · · bn (j) 0 b1 (j) b2 (j) · · · bn−1 (j) 0 0 b1 (j) · · · bn−2 (j) ... ... ... . . . ... 0 0 0 · · · b1 (j)  . We note that there is an obvious isomorphism Vn(R) ∗M ∼= Vn(R ∗M). Therefore, we can rewrite φ and ψ as φ =  φ1 φ2 φ3 · · · φn 0 φ1 φ2 · · · φn−1 0 0 φ1 · · · φn−2 ... ... ... . . . ... 0 0 0 · · · φ1  , ψ =  ψ1 ψ2 ψ3 · · · ψn 0 ψ1 ψ2 · · · ψn−1 0 0 ψ1 · · · ψn−2 ... ... ... . . . ... 0 0 0 · · · ψ1  . Let A(Vn(R))B = 0 for A = (a1, a2, . . . , an), B = (b1, b2, . . . , bn) ∈ Vn(R). For any r ∈ R,A(r, 0, . . . , 0)B = 0. Thus we have the following equations: a1 (i)rb1 (j) = 0. (2.5) a1 (i)rb2 (j) + a2 (i)rb1 (j) = 0. (2.6) a1 (i)rb3 (j) + a2 (i)rb2 (j) + a3 (i)rb1 (j) = 0. (2.7) ... a1 (i)rbn−1 (j) + a2 (i)rbn−2 (j) + · · ·+ an−1 (i)rb1 (j) = 0. (2.8) a1 (i)rbn (j) + a2 (i)rbn−1 (j) + · · ·+ an (i)rb1 (j) = 0. (2.9) Now for a monoid M and σ : M → Aut(R) a monoid homomorphism. From Eq.(2.5) we see a1(i)σgi(Rσs(b1 (j))) = 0 for all i, j and s ∈ M. Hence a1(i) ∈ ℓR(Rσs(b1 (j))). By hypothesis, R is a left APP , ℓR(Rσs(b1(j))) is left s-unital by Lemma 1. Hence there exist en ∈ ℓR(Rσs(b1 (j))) such that a1(i)en = a1 (i) since σgi is an automorphism, i = 1, 2, . . . , n. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1885 This implies that, b1(j)σhj (Rσs(a1 (i))) = 0 by rigidness and we obtain ψ1(Vn(R) ∗M)φ1 = 0. If we multiplying Eq.(2.6) on the right-hand side by tb1(j) for any t ∈ R, then a1 (i)rb2 (j)tb1 (j) + a2 (i)rb1 (j)tb1 (j) = 0. (2.10) Hence a2 (i)Rb1 (j) = 0. Since R is M -rigid we have a2 (i)σgi(Rσs(b1 (j))) = 0. Hence a2 (i) ∈ ℓR(Rσs(b1 (j))). By hypothesis, R is a left APP , then ℓR(Rσs(b1(j))) is left s-unital by Lemma 1. Hence there exist en ∈ ℓR(Rσs(b1 (j))) such that a2(i)en = a2 (i) since σgi is an automorphism. This shows that b1(j)σhj (Rσs(a2 (i))) = 0 since by rigidness we obtain ψ1(Vn(R)∗M)φ2 = 0. Thus, we deduce the other side of Eq. (2.10), a1(i)σgi(Rσs(b2 (j))) = 0 and so ψ2(Vn(R) ∗M)φ1 = 0. Similarly, if we multiply Eq.(2.7) on the right-hand side by tb1 (j) for any t ∈ R, then a1 (i)rb3 (j)tb1 (j) + a2 (i)rb2 (j)tb1 (j) + a3 (i)rb1 (j)tb1 (j) = 0. (2.11) And so a3(i)Rb1(j) = 0. Since R is M -rigid and σgi is an automorphism, we have a3 (i)σgi(Rσs(b1 (j))) = 0. Hence, a3(i) ∈ ℓR(Rσs(b1 (j))). By hypothesis, R is a left APP , ℓR(Rσs(b1 (j))) is left s-unital by Lemma 1. Hence, there exist en ∈ ℓR(Rσs(b1 (j))) such that a3(i)en = a3 (i) since by rigidness we obtain ψ1(Vn(R) ∗M)φ3 = 0. Then, Eq. (2.7) becomes a1 (i)rb3 (j) + a2 (i)rb2 (j) = 0. (2.12) If we multiplying Eq.(2.12) on the right-hand side by tb2(j) for any t ∈ R, then a1(i)rb3(j) = 0 and a2(i)rb2(j) = 0 by the similar argument to above. Thus, we have ai(i)σgi(Rσs(bj (j))) = 0 and bj(j)σhj (Rσs(ai (i)) = 0 for all 2 ≤ i+ j ≤ 4. Inductively, we assume that ai(i)σgi(Rσs(bj (j))) = 0 and bj(j)σhj (Rσs(ai (i)) = 0 for all i+ j ≤ n. if we multiply Eq.(2.9) on the right-hand side by t1b1(j), t2b2(j), . . . , tn−1bn−1 (j) for any t1, t2, . . . , tn−1 ∈ R, in turn, since R is M -rigidness and σgi is an automorphism, we have an (i)σgi(Rσs(b1 (j))) = 0, an−1 (i)σgi(Rσs(b2 (j))) = 0, . . . , a2 (i)σgi(Rσs(bn−1 (j))) = 0 and a1(i)σgi(Rσs(bn (j))) = 0. Hence, an (i) ∈ ℓR(Rσs(b1 (j))), an−1 (i) ∈ ℓR(Rσs(b2 (j))), . . . , a1 (i) ∈ ℓR(Rσs(bn (j))). By hypothesis, R is a left APP , ℓR(Rσs(b1(j))), ℓR(Rσs(b2(j))), . . . , ℓR(Rσs(bn−1 (j))) and ℓR(Rσs(bn (j))), respectively, is left s-unital by Lemma 1 again. Hence, there exist en ∈ ℓR(Rσs(b1 (j))), ℓR(Rσs(b1 (j))), . . . , ℓR(Rσs(b1 (j))) such that an(i)en = an (i), an−1 (i)en = an−1 (i), . . . , a1 (i)en = a1 (i). Now, it is straightforward to see that Aiσgi(Rσs(Bj)) = 0 for all i, j. Since R is rigidness we option Bjσhj (Rσs(Ai)) = 0. This prove that ψ(Vn(R) ∗M)φ = 0. Therefore, Vn(R) is σ-skew strongly M -reflexive. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1886 Corollary 2. (Theorem 3.8 [12]). Let R be a ring with an endomorphism α and n ≥ 2. If R is a semiprime and right α-skew reflexive ring, then Vn(R) is right α-skew reflexive. 3. Nilpotent Elements of Reflexive in Skew Monoid Rings In this section, we introduce the concept of σ-skew strongly M -nil-reflexive ring and consider its properties. Let φ = b1g1 + b2g2 + · · · + bngn ∈ R[M ]. The element φ ∈ Nil(R)[M ] if and only if bi ∈ Nil(R) for all 1 ≤ i ≤ n. Also, we say that φ ∈ Nil(R ∗M) if φ is a nilpotent element in the skew monoid ring R ∗M . For any φ ∈ R ∗M , we denote by Cφ the set of all coefficients of φ. For more details on this, please refer to [18]. Definition 3. We say that a ring R is σ-skew strongly M -nil-reflexive (σ-skew strongly nil-reflexive relative to a monoid M), if φϕψ ∈ Nil(R) ∗M implies that biσgi(c σs(aj)) ∈ nil(R), where φ = b1g1 + b2g2 + · · · + bngn, ϕ = c1l1 + c2l2 + · · · + cdld and ψ = a1h1 + a2h2 + · · ·+ amhm ∈ R ∗M, then ψϕφ ∈ Nil(R) ∗M for all i, k, j. If M = (N ∪ {0},+) and σg = idR for all g ∈ M , then a ring R is σ-skew strongly M -nil-reflexive if and only if R is strongly nil-reflexive. Also, if M = {e} and σg = idR for all g ∈M, then any ring R is σ-skew strongly M -nil-reflexive. A ring R is called an NI ring if nil(R) forms an ideal. For a unique product monoid we have the following result. Theorem 3. Let R be an NI-ring, M be a u.p.-monoid, and σ : M → Aut(R) be a compatible monoid homomorphism, then R is σ-skew strongly M -nil-reflexive Proof. Let φ = b1g1 + b2g2 + · · · + bngn, ϕ = c1l1 + c2l2 + · · · + cdld and ψ = a1h1 + a2h2 + · · · + amhm be nonzero elements in R ∗M such that φϕψ ∈ nil(R) ∗M implies that biσgi(σs(c aj)) ∈ nil(R). We will show that ajσhj (σs(c bi)) ∈ nil(R) for each 1 ≤ i ≤ n, 1 ≤ k ≤ d and 1 ≤ j ≤ m. We proceed by induction on both n and m. By using freely σ is compatible monoid homomorphism. Let n = 1 and hence φ = b1g1. Since M is u.p.- monoid then by Lemma 4, M is cancellative monoid, g1hi ̸= g1hj for i ̸= j. So b1σg1(c(aj)) ∈ nil(R) for any c ∈ R and each j. The proof of the case m = 1 is similar. Now, let m,n ≥ 2. Since M is u.p.-monoid, there exist i, j with 1 ≤ i ≤ n and 1 ≤ j ≤ m such that gishj is uniquely presented by considering two subsets K = {g1, g2, . . . , gn} and H = {sh1, sh2, . . . , shm} of a monoid M, s ∈ M. We may assume without loss of generality, that i = n and j = m. Thus bnσgn(c (am)) ∈ Nil(R). Hence for any c ∈ R, bnσgn(ck(am)) ∈ nil(R) so, there exist a positive integer k ≥ 1 such that (bnσgn(c (am))) k = 0. Then (bnσgn(c (am)))(bnσgn(c (am))) · · · (bnσgn(c (am))) = 0. Now, since by hypothesis σ :M → Aut(R) is a compatible monoid homomorphism, we have that (bnσgn(c am)) · · · (bnσgn(c am))(bnc am) = 0 and then we conclude that (bnσgn(c am)) · · · (bnσgn(c am))σgn(bnc am) = 0. So (bnσgn(c am)) · · · bnσgn(c ambnam) = 0 and hence (bnσgn(c am)) · · · (bnσgn(c am))(bnc am)2 = 0. Continuing this procedure yields that (bnc am) k = 0, hence bnσgn(c am) ∈ Nil(R). Therefore, anσgn(σs(c bm)) ∈ nil(R) for each c ∈ R. Then we have am(φ − bngn)ϕψ = amφϕψ − ambngnϕψ ∈ nil(R) ∗M, since nil(R) E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1887 is an ideal of R. By induction hypothesis and by compatibility, we have bnσgn(c ajbn) = 0 implies biσgn(c am) ∈ nil(R) for all i. Applying the preceding method repeatedly, we obtain biσgi(σs(c aj)) ∈ nil(R) for each i, j and each c ∈ R. Thus, by rigidness we have ajσhj (σs(c bi)) ∈ nil(R) as desired. Corollary 3. (1) Every σ-compatible NI-ring is σ-skew strongly Z-nil-reflexive. (2) Every NI-ring is nil-reflexive. Proof. (1) Taking M = {. . . , x−2, x−1, 1, x1, x2, . . .} and σxn(λ) = σn(λ) for each n ∈ Z and any λ ∈ R, we have R ∗M ∼= R[x, x−1, σ], and the result follows from Theorem 3. (2) Taking M = {1, x1, x2, . . .} and σxn(λ) = λ for each n ∈ N ∪ {0} and any r ∈ R, we have R ∗M ∼= R[x], and the result follows from Theorem 3. Corollary 4. (Theorem 3.1 [20]) Let M be a u.p.-monoid and R be a reduced ring. Then, R is strongly M -reflexive. Corollary 5. Let R be a ring, where nil(R) is an ideal of R. Let M be a u.p.-monoid and σ :M → Aut(R) be a compatible monoid homomorphism. Then, R is σ-skew strongly M -nil-reflexive Proof. Since nil(R) is an ideal, it is an NI-ring. Therefore, the proof follows from Theorem 3. The following example shows that R being a reduced ring in Theorem 3 is not super- fluous. Example 2. Let M be a monoid with | M |≥ 2, S = M2(F ) and σ : M → Aut(R) be a compatible monoid homomorphism. Then, S is not σ-skew strongly M -nil-reflexive. Proof. Take e ̸= g ∈M and let φ = ( 0 1 0 0 ) e+ ( 1 0 0 0 ) g, ψ = ( 1 1 0 0 ) e+ ( 1 1 0 0 ) g ∈ S ∗M. For ϕ = ( 1 0 0 1 ) g ∈ S ∗M, it is easy to check that φϕψ ∈ nil(S ∗M). But, we have ψϕφ = ( 0 1 0 0 ) g + ( 1 0 0 −1 ) g2 + ( 0 0 −1 0 ) g3 ̸= 0 which implies that S is not σ-skew strongly M -nil-reflexive. Shaban and Mohammed [22] proved that Nil(R ∗M) ̸= Nil(R) ∗M for an NI-ring R and a u.p.-monoid M with |M |≥ 2. Based on their work, we have the following result. Theorem 4. Let R be any ring, M be any monoid with | M |≥ 2 and σ : M → Aut(R) be a compatible monoid homomorphism such that Nil(R ∗M) = Nil(R) ∗M. Then R is σ-skew strongly M -nil-reflexive. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1888 Proof. Let φ = Σni=1bigi and ψ = Σmj=1ajhj be nonzero elements in R ∗M such that φϕψ ∈ nil(R) ∗M for any ϕ ∈ R ∗M. We have φϕψ ∈ nil(R ∗M), which is equivalent to the existence of a positive integer ℓ such that (φϕψ)ℓ = (φϕψ)(φϕψ) . . . (φϕψ) = 0. Since σ is a compatible monoid homomorphism, we have biσgi(c(aj)) ∈ nil(R). Therefore, ajσhj (c(bi)) ∈ nil(R), and the proof is complete. In Proposition 3.4 [20], it was proved that if I is a reduced ideal and R/I is strongly M -reflexive, then R is strongly M -reflexive. Based on this result, we have the following statement. Theorem 5. Suppose that R is a ring, M is a strictly ordered monoid, and σ : M → Aut(R) is a compatible monoid homomorphism. If I is an ideal of R contained in nil(R), then R/I is σ-skew strongly M -nil-reflexive if and only if R is σ-skew strongly M -nil- reflexive. Proof. “ =⇒ ” Let φ,ψ ∈ R∗M satisfying φϕψ ∈ nil(R)∗M for all ϕ ∈ R∗M. We write φ = b1g1+ b2g2+ · · ·+ bngn, ϕ = c1l1+ c2l2+ · · ·+ cdld and ψ = a1h1+ a2h2+ · · ·+ amhm with g1 < g2 < . . . < gn, h1 < h2 < . . . < hm. We will use transfinite induction on the strictly totally ordered set (M,≤) to show that ajσhj (σs(r bi)) ∈ nil(R). Since R/I is σ-skew strongly M -nil-reflexive and 0̄ = (b̄1g1 + b̄2g2 + · · ·+ b̄ngn)(rs)(ā1h1 + ā2h2 + · · ·+ āmhm) = (b1 + I)σ̄g1(r(a1 + I)) + (b2 + I)σ̄g2(r(a2 + I)) + · · ·+ (bn + I)σ̄gn(r(am + I)) = (b1σ̄g1(r a1) + I) + (b2σ̄g2(r a2) + I) + · · ·+ (bnσ̄gn(r am) + I) ∈ (R/I) ∗M for r ∈ R, s ∈ M, so we have biσgi(r aj) ∈ I for all i, j. Since M is a strictly totally ordered monoid, we have g1h1 < gih1 ≤ gihj = g1h1 for i ̸= 1 or j ̸= 1. It follows that b1σg1(r a1) = 0, i.e., b1σg1(σs(r a1)) ∈ nil(R) since I is an ideal of R contained in nil(R). Now suppose that bir aj = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ m with w ∈ M is such that for any gi and hj , gihj < w. We will show that biσgi(σs(raj)) ∈ nil(R) for any gi and hj with gihj = w. Set X = {(gi, hj)|gihj = w}. Then X is a finite set. We write X as {(git , hjt)|t = 1, 2, . . . , k} such that gi1 < gi2 < . . . < gik . Since M is cancellative, gi1 = gi2 and gi1hj1 = gi2hj2 = w imply hj1 = hj2 . Since ≤ is a strict order, gi1 < gi2 and gi1hj1 = gi2hj2 = w imply hj2 < hj1 . Thus, we have hjk < hjk−1 < . . . < hj2 < hj1 . Now ∑ (gi,hj)∈X biσgi(σs(raj)) = k∑ t=1 bitσgit (σs(rajt)) = 0. For any t ≥ 2, gi1hjt < githjt = w, and so bi1σgit (σs(rajt)) = 0 by induction hypothesis. Thus, bi1rajt = 0 because R is M -compatible. Since I is reduced and σgit (ajt)I(bi1) ⊆ I, then we have (ajtIbi1) 2 = 0 and I is reduced. Thus, for any t ≥ 2, (bitrajt)(bi1raj1) 2 = (bitrajt)(bi1raj1)(bi1raj1) ∈ (bitrajt)I(bitrajt) = (bitrajt)I(bitrajt) = 0, which implies that (bitrajt)(bi1raj1) 2 = 0. Now multiplying ∑k t=1 bitσit(σs(r ajt)) = 0 on the right by E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1889 (bi1σi1(raj1)) 2, we obtain 0 = (bitσit(rajt))(bi1σi1(raj1)) 2 = (bi1σi1(raj1)) 2(bi1σi1(raj1)) = (bi1σi1(raj1)) 3. Since bi1σi1(raj1) ⊆ I and I is reduced and R is M -compatible, we have bi1σi1(raj1) = 0. Thus, ∑k t=2 bitσit(rajt) = 0. Then, bitσit(rajt) ∈ nil(R) for t ≥ 2.Multiplying (bi2σi2(raj2)) 2 on ∑k t=2 bitσit(rajt) = 0 from the right-hand side, we obtain bi2σi2(raj2) = 0. Thus, bi2σi2(raj2) ∈ nil(R) by the same way as the above. Continuing this process, we can prove bitσit(rajt) = 0 for t = 1, . . . , k. Thus biσgi(σs(raj)) ∈ nil(R) because I is an ideal of R contained in nil(R) for any i and j with gihj = w. Therefore, by transfinite induction biσgi(σs(raj)) ∈ nil(R) for any i and j. Thus, ajσhj (σs(rbi)) ∈ nil(R). Therefore, R is σ-skew strongly M -nil-reflexive. “ ⇐= ” Let φ̄, ψ̄ ∈ (R/I) ∗M with φ̄ϕ̄ψ̄ ∈ nil(R/I) ∗M for all ϕ̄ ∈ (R/I) ∗M. Where φ = b1g1+b2g2+ · · ·+bngn, ϕ = c1l1+c2l2+ · · ·+cdld and ψ = a1h1+a2h2+ · · ·+amhm be nonzero elements in R∗M. Since Nil(R) is an ideal of R, we write R = R/nil(R) and define σ :M → Aut(R) by σg(x+Nil(R)) = σg(x)+Nil(R), for each g ∈M. We claim that σ is a compatible monoid homomorphism. For each x, y ∈ R, let (x+Nil(R))(y+Nil(R)) = 0̄. Then xy ∈ Nil(R) and hence (x + Nil(R))σ̄(y + Nil(R)) = 0̄. The converse is similar. Therefore, σ is compatible. For any φ = b1g1 + b2g2 + · · · + bngn ∈ R ∗M, we denote φ̄ = ∑n i=1(bi+ nil(R))gi ∈ R̄ ∗M. It is easy to see that the mapping σ : R ∗M −→ R ∗M defined by σ(φi) = φi is a ring homomorphism. Since φ̄ϕ̄ψ̄ ∈ nil(R/I) ∗M. Then there exist a positive integer ℓ ∈ N such that (φ̄ϕ̄ψ̄)ℓ ∈ nil(R/I) ∗M. So biσgi(raj)ℓ ∈ I for any i, j and r ∈ R. So biσgi(σs(raj)) ∈ nil(R) since I ⊆ nil(R) and by compatibility, we have biσgi(raj) ∈ nil(R) and ajσhj (σs(rbi)) ∈ nil(R) since R is σ-skew strongly M -nil-reflexive, this mean that ψϕφ ∈ nil(R) ∗M. Thus ψ̄ϕ̄φ̄ ∈ nil(R/I) ∗M. Therefore, R/I is σ̄-skew strongly M -nil-reflexive. Proposition 3. Let M be a finitely generated abelian group and σ : M → Aut(R) be a compatible monoid homomorphism. Then M is torsion free if and only if there exist a non-zero ring R such that R is σ-skew strongly M -nil-reflexive. Proof. Let M be a finitely generated torsion-free abelian group. Thus, M ∼= ⨿N1 Z and so M is a u.p.-monoid, as shown in Lemma 2. Therefore, for any NI ring R, R is σ-skew strongly M -nil-reflexive by Theorem 3. Conversely, suppose gℓ = e for some e ̸= g ∈ M and positive integer ℓ. Let N be a cyclic subgroup of M generated by {g}. Therefore, R is σ-skew strongly N -nil-reflexive, which leads to a contradiction, as shown in Lemma 3. Hence, M is torsion-free. We will now provide some examples of σ-skew strongly M -nil-reflexive ring. In Theo- rem 2.6 [13], Kwak and Lee proved that R is a reflexive ring if and only if Matn(R) is a reflexive ring for all n ≥ 1. However, this is not the case in σ-skew strongly M -nil-reflexive rings of R. There exist σ-skew strongly M -nil-reflexive rings over which matrix rings need not be σ-skew strongly M -reflexive, as shown below. E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1890 Example 3. Let S be a torsion-free and cancellative monoid and σ : M → Aut(R) be a compatible monoid homomorphism. (1) If R is a ring with nil(R) an ideal of R, then R is σ-skew strongly M -nil-reflexive. (2) For any reduced ring R, the ring Tn(R) is σ-skew strongly M -nil-reflexive. However, the ring of all 2× 2 matrices over any field is not σ-skew strongly M -nil-reflexive. (3) For R be a reduced ring. Consider the ring Sn(R) =   a a12 a13 · · · a1n 0 a a23 · · · a2n 0 0 a · · · a3n ... ... ... . . . ... 0 0 0 · · · a  | a, aij ∈ R; 1 ≤ i, j ≤ n  . Then Sn(R) is not σ-skew strongly M -reflexive, when n ≥ 4, but Sn(R) and R are σ-skew strongly M -nil-reflexive for all n ≥ 1. Solution (1). Suppose φ,ψ ∈ R ∗M, with φϕψ is nilpotent for all ϕ ∈ R ∗M , where φ = b1g1+ b2g2+ · · ·+ bngn, ϕ = c1l1+ c2l2+ · · ·+ cdld and ψ = a1h1+a2h2+ · · ·+amhm. So there exist a positive integer ℓ such that (φϕψ)ℓ = 0. Therefore (biσi(σs(caj))) ℓ = 0, for any s ∈ M, i, j. Then, biσgi(σs(c aj)) ∈ nil(R) and so ajσhj (σs(c bi)) ∈ nil(R). Hence ψϕφ is nilpotent. (2). Let R be a ring, by [4], nil(Tn(R)) =  nil(R) R R · · · R 0 nil(R) R · · · R 0 0 nil(R) · · · R ... ... ... . . . ... 0 0 0 · · · nil(R)  . Assuming that R is a reduced, we know that nil(R) = 0 and that nil(Tn(R)) is an ideal. From (1), it follows that Tn(R) is σ-skew strongly M -nil-reflexive. However, if we take A = E12 and B = E22 ∈Mat2(F ), where F is a field, and C ∈Mat2(F ), we see that ACB is nilpotent, but BCA = E22 is not nilpotent. This shows that Mat2(F ) is not σ̄-skew strongly M -nil-reflexive. (3). By the same argument as in Example 2.3 [13]. For a nonzero reduced ring S, the ring R =   α β δ 0 α γ 0 0 α  | α, β, γ ∈ S  is semicommutative by Proposition 1.2 [11]. If we take φ = CE23g and ψ = CE12h ∈ Sn(R)∗M for any g, h ∈M , we have φ(Sn(R)∗M)ψ = 0 but ψ(Sn(R) ∗M)φ ̸= 0. Therefore, Sn(R) is not σ-skew strongly M -reflexive. Since R is E. Ali / Eur. J. Pure Appl. Math, 16 (3) (2023), 1878-1893 1891 reduced, it follows that Sn(R) is σ-skew strongly M -nil-reflexive. Note that nil(Sn(R)) =   a a12 a13 · · · a1n 0 a a23 · · · a2n 0 0 a · · · a3n ... ... ... . . . ... 0 0 0 · · · a  | a ∈ nil(R), aij ∈ R; 1 ≤ i, j ≤ n  . The ring R being reduced implies that nil(Sn(R)) is an ideal. By (1), Sn(R) is σ-skew strongly M -nil-reflexive. By Example 3(2) for n by n upper triangular matrix ring over R. It is easy to verify the next result. Proposition 4. Let M be a torsion-free and cancellative monoid and σ : M → Aut(R) a compatible monoid homomorphism. A ring R is σ-skew strongly M -nil-reflexive if and only if Tn(R) is σ-skew strongly M -nil-reflexive, for any positive integer n. Proof. It suffices to show “ ⇒ ” Let φ,ψ ∈ Tn(R) ∗M such that φϕψ ∈ nil(Tn(R)) ∗ M , where φ = (bij), ϕ = (cij) and ψ = (aij) for all (i, j)th entry of the matrix. Since nil(Tn(R)) = {(aij)|aij ∈ nil(R)}, then we have Cφiiϕiiψii ∈ nil(R) for each 1 ≤ i ≤ n. Since R is σ-skew strongly M -nil-reflexive, there exist some positive integer mi such that (biiσgii(σs(ciiaii))) mi = 0. Then, by compatibility biiσgii(ciiaii) is nilpotent and so aiiσhii(σs(ciibii is nilpotent. Thus, ψϕφ ∈ nil(Tn(R)) ∗ M. Therefore, Tn(R) is σ-skew strongly M -nil-reflexive. Proposition 5. Let M be a strictly ordered monoid and σ : M → Aut(R) a compatible monoid homomorphism. If R is finite subdirect product of σ-skew strongly M -nil-reflexive rings, then R is σ-skew strongly M -nil-reflexive. Proof. Let Ik(k = 1, . . . , l) be ideals of R such that R/Ik is σ-skew strongly M -nil- reflexive and ⋂l k=1 Ik = 0. Let φ and ψ be elements in R ∗M such that φϕψ ∈ nil(R) ∗M for all ϕ ∈ R ∗M . Clearly, φ̄ϕ̄ψ̄ ∈ nil(R/Ik) ∗M . Since R/Ik is σ-skew strongly M - nil-reflexive, we have (biσgi(σs(raj))) ℓ ∈ Ik, where σ is compatible and r ∈ R, for some positive integer ℓ. Therefore, (biσi(σs(raj)))ℓ ∈ ⋂l k=1 Ik = 0. Hence, (biσgi(σs(raj)))ℓ = 0, and we conclude that biσgi(σs(raj)) ∈ nil(R) since σ is compatible and nil-reflexivity. Then, ajσhj (σs(rbi)) ∈ nil(R). Therefore, ψϕφ ∈ nil(R) ∗M, the proof is done. 4. Conclusion This paper introduced and studied two important concepts, namely σ-skew strongly M -reflexive and σ-skew strongly M -nil-reflexive. 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