EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1637-1648 ISSN 1307-5543 – ejpam.com Published by New York Business Global Involution t – Clean Rings with Applications Shaimaa H. Ahmad1, Mohammed Al-Neima2,∗, Ahmed Ali1, Raida D. Mahmood1 1 Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq 2 Department of Civil Engineering, College of Engineering, University of Mosul, Mosul, Iraq Abstract. A new class of rings was introduced, in which every element in the ring is a sum of in- volution and tripotent elements. This class called involution t-clean rings, which is a generalization of invo-clean rings and subclass of clean rings. Some properties of this class are investigated. For an application in graph theory, a new graph is defined as t-clean graph of involution t-clean ring. The set of vertices is ordered pairs of involution and tripotent element, which is the sum of them is involution t-clean element. The two vertices are adjacent if and only if the sum of involution elements is zero or the product of the tripotent elements is zero. The graphs are connecting, has diameter one and girth three. 2020 Mathematics Subject Classifications: 16E50, 16U99, 05C25 Key Words and Phrases: Clean ring, invo-clean ring, tripotent element, Hosoya polynomial, Wiener index 1. Introduction Throughout this paper, the ring R is associative with identity. The symbols U(R) is the set of unit elements in R, Idm(R) is the set of idempotent elements in R, invo(R) the subset of U(R) consisting of all involution elements of R, Tri(R) = {t ∈ R : t3 = t}, N2(R) = {x ∈ R : x2 = 0}. The aim of this paper is to generalize the concept of invo-clean ring,the purpose to obtain more general results by generalizing some results of invo-clean ring. This prompts researchers to study many of the properties of this generalization. The concept of clean ring, first defined by Nicholson in 1977 [9], a ring R is called clean, if for all x ∈ R and written as x = u + e, where u ∈ U(R) and e ∈ Idm(R). If ue = eu, get that R is called strongly clean. Recall that a ring R is called invo-clean, if every x ∈ R can be written as x = u+ e, where u ∈ invo(R) and e ∈ Idm(R). If, ue = eu, get that R is strongly invo-clean.Which was first defined Danchev see [6]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4530 Email addresses: shaymaahatim@uomosul.edu.iq (Shaimaa H. Ahmad), mohammedmth@uomosul.edu.iq (Mohammed Al-Neima), ahmedgraph@uomosul.edu.iq (Ahmed Ali), raida.1961@uomosul.edu.iq (Raida D. Mahmood) https://www.ejpam.com 1637 © 2022 EJPAM All rights reserved. Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1638 Every idempotent element is invo-clean, because e = (2e−1)+(1−e), where (2e−1)2 = 1 and (1− e)2 = 1− e. Moreover, Z2, Z3, Z4, Z6, Z8 are invo-clean rings. Every invo-clean ring is clean, but the opposite is untrue. (Examples Z5, Z10). As usual, M2(R) stand for the 2×2 matrix ring. T2(R) is 2×2 upper triangular matrix ring. We write Zn for rings of integers modulo n. In [11], the ring R is regular if every a ∈ R, a ∈ aRa. The ring R is called r-clean if all elements of R can be written as the summation of a regular and idempotent elements [2]. Zero divisor graph is concept give the relation between the commutative rings and graph theory, which was first defined by Beck in [4]. In the same way, Habibi, Celikel and Abdioglu in [8] start study of clean graph defined in clean ring. For an application to invo-t-clean ring, a new graph is defined; depended on an involution t-clean ring. In graph theory, the graph is an order pair denoted by G = (V, F ) of two sets V (is non-empty) and F (may be empty) such that F ⊆ V × V .G ′ = (V ′ , F ′ ) is a sub-graph of G = (V, F ) written as G ′ ⊆ G if V ′ ⊆ V and F ′ ⊆ F . The order of G is the number of vertices, denoted by |G| and the number of edges is denoted by ∥G∥. The girth is the length of the shortest cycle in the graph G, denoted by g(G). Each edge incident on two vertices, the number of the edges incident on a vertex v is said to be the degree of v and it’s denoted by degG(v) or deg(V ). The numbers δ(G) and ∆(G) are minimum and maximum degrees r of a graph G respectively [5]. The average degree of G is defined by: ad(G) = 1 |G| ∑ ∀v∈V deg(v). From clearly that δ(G) ≤ ad(G) ≤ ∆(G). The distance d(v, u) in connected graph G is positive number represent. The length shortest (v − u)- path in G. The maximum distance between any two distinct vertices in G is the diameter of G and is denoted by diam(G). The sum of the lengths of the shortest (v − u)- path in G is called Wiener index [12], that is W (G) = 1 2 ∑ ∀v,u∈V d(v, u). The average distance is define by: D(G) = 2W (G) |G|(|G|−1) . Let d(G, k) be the number of vertex pairs at distance k in a connected graph G, where k = 0, 1, · · · , diam(G). Then the Hosoya polynomial [7] of G is defined as: H(G, t) = ∑diam(G) k=1 d(G, k)tk. From clearly that: W (G) = d dtH(G, t)|t=1 = ∑diam(G) k=0 kd(G, k).∑diam(G) k=0 d(G, k) = |G|(|G|+1) 2∑diam(G) k=1 d(G, k) = |G|(|G|−1) 2 ad(G) = 2∥G∥ |G| , d(G, 0) = |G| andd(G, 0) = ∥G∥. Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1639 2. Invo-t-clean Rings The section describes the structure of the invo-t-clean rings. Definition 2.1. The ring R is called involution t-clean (for short invo-t-clean), if all a ∈ R can be expressed as a = u+ t, where u ∈ invo(R) and t ∈ Tri(R). An invo-t-clean ring with ut = tu is strongly invo-t-clean. Example 2.2. 1. Let n ≥ 2 and integer. Then the ring Zn is invo-t-clean if and only if n = 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. 2. Let M2(Z2),M2(Z3) and the upper triangular matrices T2(Z2), T2(Z3) are invo-t- clean rings. Remark 2.3. 1. Every invo-t-clean ring is also clean, but the reverse is not true, for a commutative example Z7. For non-commutative example M2(Z4), T2(Z4). The elements [ 2 3 0 2 ] , [ 2 1 0 2 ] , [ 0 1 0 0 ] , [ 0 3 0 0 ] are not invo-t-clean in, T2(Z4), where U2(T2(Z4)) =  [ 1 0 0 1 ] , [ 1 0 0 3 ] , [ 1 1 0 3 ] , [ 1 2 0 1 ] , [ 1 2 0 3 ] , [ 1 3 0 3 ] ,[ 3 0 0 1 ] , [ 3 0 0 3 ] , [ 3 1 0 1 ] , [ 3 2 0 1 ] , [ 3 2 0 3 ] , [ 3 3 0 1 ]  Tri(T2(Z4)) =  [ 0 0 0 0 ] , [ 0 0 0 1 ] , [ 0 0 0 3 ] , [ 0 1 0 1 ] , [ 0 1 0 3 ] , [ 0 2 0 1 ] ,[ 0 2 0 3 ] , [ 0 3 0 1 ] , [ 0 3 0 3 ] , [ 1 0 0 0 ] , [ 1 0 0 1 ] , [ 1 0 0 3 ] ,[ 1 1 0 0 ] , [ 1 1 0 3 ] , [ 1 2 0 0 ] , [ 1 2 0 1 ] , [ 1 2 0 3 ] , [ 1 3 0 0 ] ,[ 1 3 0 3 ] , [ 3 0 0 0 ] , [ 3 0 0 1 ] , [ 3 0 0 3 ] , [ 3 1 0 0 ] , [ 3 1 0 1 ] ,[ 3 2 0 0 ] , [ 3 2 0 1 ] , [ 3 2 0 3 ] , [ 3 3 0 0 ] , [ 3 3 0 1 ]  2. Every invo-clean ring is invo-t-clean, but the reverse is not true. For commuta- tive example Z5, the element 3 is not invo-clean. For non-commutative example M2(Z3), T2(Z3). The elements [ 0 2 0 0 ] , [ 0 1 0 0 ] , [ 1 2 0 1 ] , [ 1 1 0 1 ] are not invo-clean in T2(Z3), where U2(T2(Z3)) =  [ 1 0 0 1 ] , [ 1 0 0 2 ] , [ 1 1 0 2 ] , [ 1 2 0 2 ] ,[ 2 0 0 1 ] , [ 2 0 0 2 ] , [ 2 1 0 1 ] , [ 2 2 0 1 ]  Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1640 Id(T2(Z3)) =  [ 0 0 0 0 ] , [ 0 0 0 1 ] , [ 0 1 0 1 ] , [ 0 2 0 1 ] ,[ 1 0 0 0 ] , [ 1 0 0 1 ] , [ 1 1 0 0 ] , [ 1 2 0 0 ]  We begin with two useful technicalities. Proposition 2.4. Let 3 be an invo-t-clean element in a ring R. Then 15t = 0 and 120 = 0. Proof. Since 3 = u+ t, where u ∈ invo(R) and t ∈ Tri(R).Thus ut = (3− t)t = 3t− t2 = t(3− t) = tu, so ut = tu. Now 24 = 33 − 3 = (u+ t)3 − (u+ t) = 8 · 3 = 8(u+ t), u3 + 3u2t+ 3ut2 + t3 − ut = 8(u+ t) u+ 3t+ 3ut2 + t− u− t = 8(u+ t). −3t− 3ut2 + 8u+ 8t = 0. Multiply both sided from the right by t −3t2 − 3ut3 + 8ut+ 8t2 = 5t(t+ u) = 0. Hence 15t = 0 In addition, 24 = 3t+ 3ut2 = 3t+ 3(3− t)t2 = 3t+ 9t2 − 3t3 = 9t2. So 120 = 5 · 24 = 5 · 9t2 = 3(15t)t = 0.■ Proposition 2.5. The homomorphic image of invo-t-clean rings are invo-t-clean. Proof. Let f : R −→ S be a homomorphic mapping from an invo-t-clean ring R into S. Then for all y ∈ S there x ∈ R with y = f(x) and x = u + t where u ∈ invo(R) and t ∈ Tri(R). Now y = f(x) = f(u+t) = f(u)+f(t), (f(t))3 = f(t3) = f(t) ∈ Tri(R). Since u ∈ U2(R),then (f(u))2 = f(u2) = f(1) = 1, so f(u) ∈ U2(R) y = f(x) = f(u) + f(t). Therefore S is invo-t-clean ring. ■ Corollary 2.6. Let R be an invo-t-clean ring and I is an ideal of R. Then R/I is an invo-t-clean. Since the involution element is also a tripotent element. The following proposition explains that the nilpotent element of index 2 will be a sum of two involution elements if it is invo-t-clean. Proposition 2.7. For a ring R. Let a ∈ N2(R). 1. If a is an invo-t-clean element, then t ∈ invo(R). 2. If a is a strongly invo-t-clean element, then 2a = 0. Proof. (1) Since a ∈ N2(R), then a2 = 0 and a = u + t, where u ∈ invo(R) and t ∈ Tri(R). 0 = a2 = (u+ t)2 = 1+ut+ tu+ t2. Multiply both sided from the right and left by t. 2t2 + tut2 + t2ut = 0 Since a2 = 0, then 0 = (ua)2u+ ta2t = u(u+ t)2u+ t(u+ t)2u = 1+ tu+ ut+ ut2u+ t2 + tut2 + t2ut+ t2 = −t2 + ut2u. So t2 = ut2u implies ut2 = t2u. 1 + ut+ tu+ t2 = 0 = 2t2 + tut2 + t2ut = 2t2 + tu+ ut t2 = 1. (2): 0 = a2 = (u + t)2 = 1 + 2ut + t2. Multiply both sided from the right by t, 0 = 2ut2 + 2t = 2t(u+ t) = 2ta = 2t2a = 2a from(1). ■ Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1641 Remark 2.8. The invo-t-clean elements in N2(Zn) is either 0 or 0, n2 .For example 1. For an invo-t-clean ring take Z12, then every elements in N2(Z12) = 0, 6 is invo- t-clean where 6 = 1 + 5 = 7 + 11, where invo(Z12) = {1, 5, 7, 11}, Tri(Z12) = {0, 1, 3, 4, 5, 7, 8, 9, 11}. 2. For a ring which is not invo-t-clean take Z36, N2(Z36) = 0, 6, 12, 24, 30 the only invo-t-clean elements in N2(Z36) is 0, 18, 18 = 1+ 17 = 19+ 35, where invo(Z36) = {1, 17, 19, 35}, Tri(Z36) = {0, 1, 8, 9, 17, 19, 27, 28, 35}. In noncommutative ring R, the invo-t-clean elements in N2(R) need not strongly invo- t-clean elements, but it is still satisfies the proposition 2.7, which every elements in N2(R) is sum of two involution elements, for example the ring T2(Z3) N2(T2(Z3)) = {[ 0 0 0 0 ] , [ 0 1 0 0 ] , [ 0 2 0 1 ]} where [ 0 0 0 0 ] = [ 1 0 0 1 ] + [ 2 0 0 2 ] , [ 0 1 0 0 ] =[ 1 1 0 2 ] + [ 2 0 0 1 ] , [ 0 2 0 0 ] = [ 1 2 0 2 ] + [ 2 0 0 1 ] . Proposition 2.9. Let a is a strongly invo-t-clean element in R. Then 1. −a is strongly invo-t-clean element. 2. au is strongly invo-t-clean element. 3. at is a sum of idempotent and tripotent elements. 4. If 2 is invertible. Then a2 − 1 is r-clean element. 5. If char(R) = 4. Then a = (u− 2t) + 3t, u− 2t ∈ invo(R) and 3t ∈ Tri(R). 6. If char(R) = 6. Then a2 ∈ Idm(R). Proof. (1), (2) and (3) is clearly. (4) since a is strongly invo-t-clean element, then there exists u ∈ invo(R) and t ∈ Tri(R) such that a = u+ t and ut = tu, a2 − 1 = (u+ t)2 − 1 = 1 + 2ut+ t2 − 1 = 2ut+ t2. For proving 2ut is regular. Set b = ut2−1, (2ut)b(2ut) = (2ut)(ut2−1)(2ut) = 2ututut = 2ut. Therefore, a2−1 is a r-clean element. (5) since a = u + t = (u − 2t) + 3t, (u − 2t)2 = u2 − 4ut + 4t2 = 1, u − 2t ∈ invo(R), (3t)3 = 27t3 = 3t ∈ Tri(R). (6) since a = u + t, a − t = u, (a − t)2 = u2 = 1, a2 − 2at + t2 = 1, multiply by t, a2t − 2at2 + t3 = t, a2t = 2at2 multiply by t , a2t2 = 2at, a2 − 2at = 1 − t2, (a2 − 2at)2 = (1 − t2)2, a4 − 4a3t + 4a2t2 = 1 − t2, a4 − 4a(t(a)2t2) + 4a2t2 = 1 − t2, a4 − 4at(2at) + 4(2at) = 1 − t2, a4 − 8a2t2 + 8at = 1 − t2, a4 − 16at + 8at = a2 − 2at, a4 − 6at = a2, soa4 = a2. ■ Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1642 Proposition 2.10. If R is a ring with J(R) is strongly invo-t-clean. Then J(R) is nil with index of nilpotent at most 3, and char(J(R)) = 4. Proof. Let a ∈ J(R), since J(R) is strongly invo-t-clean, then there exists u ∈ invo(R) and t ∈ Tri(R) such that a = u + t, −u + a = t ∈ U(R) + J(R) = U(R), so t ∈ U(R) ∩ J(R) = U2(R), t2 = 1. Now a2 = (u+ t)2 = 2+2ut = 2u(u+ t) = 2ua, multiply a2 = 2ua both side by a, a3 = 2ua2 = 2u(2ua) = 4a, replacing a by 2a, (8a)3 = 8a, 8a(1− a2) = 0, since a ∈ J(R), 1−a2 ∈ U(R), get that 8a = 0. For a2 = 2ua replacing a by 2a, 4a2 = 4ua. by multiply a2 = 2ua by 4 get (4a)2 = 8ua. So 8ua = 4ua, 4ua = 0, 4a = 0. a3 = 4a = 0.■ 3. Applications in Graph Theory Definition 3.1. The graph of invo-t-clean ring R, which is denoted by Clt(R) has vertex set V (Clt(R)) = {(u, t) : u ∈ invo(R)&t ∈ Tri(R)} and has the edge set F (Clt(R)) = {h1h2 : h1 = (u1, t1), h2 = (u2, t2),u1+u2 = 0 or t1 ·t2 = 0, ui ∈ invo(R), ti ∈ Tri(R), ui + ti is an invo-t-clean element i = 1, 2 }. For example, let V (Clt(Z4)) = {[1, 0], [3, 0], [1, 1], [1, 3], [3, 1], [3, 3]} Then, F (Clt(Z4)) = {[1, 0][1, 0], [1, 0][3, 0], [1, 0][1, 1], [1, 0][3, 1], [1, 0][1, 3], [1, 0][3, 3], [3, 0][3, 0], [3, 0][1, 1], [3, 0][3, 1], [3, 0][1, 3], [3, 0][3, 3], [1, 1][3, 1], [1, 1][3, 3], [1, 3][3, 1], [1, 3][3, 3]}. From the above example, we see that there is a loop at vertices [1, 0] and [3, 0] and because it is not important in graph theory, we will ignore it, and the girth of Clt(Z4) is three. There are many recent studies that connected the ring theory with the graph theory to review the paper, see [3] [10] [1]. Using the Python Programming Language, we get all connected graphs and the result from definition 3.1, and then we get some invariant properties in graph theory, such as: Hosoya polynomial, Wiener index, average distance and average degree. In the next table, the graphs structures corresponding invo-t-clean ring Zn, n = 2, 3,4, 5, 6,8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Table 1: The graphs Clt(Zn). Ring Graph Z2 H(Clt(Z2);x) = 2 + x, W (Clt(Z2)) = 1, ad(Clt(Z2)) = 1D(Clt(Z2)) = 1 Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1643 Ring Graph Z3 H(Clt(Z3);x) = 6 + 13x + 2x2, W (Clt(Z3)) = 17, ad(Clt(Z3)) = 4.333, D(Clt(Z3)) = 1.1333 Z4 H(Clt(Z4);x) = 6 + 13x + 2x2, W (Clt(Z4)) = 17, ad(Clt(Z4)) = 4.333, D(Clt(Z4)) = 1.1333 Z5 H(Clt(Z5);x) = 6 + 13x + 2x2, W (Clt(Z5)) = 17, ad(Clt(Z5)) = 4.333, D(Clt(Z5)) = 1.1333 Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1644 Ring Graph Z6 H(Clt(Z6);x) = 12 + 30x + 16x2, W (Clt(Z6)) = 82, ad(Clt(Z6)) = 5, D(Clt(Z6)) = 1.242 Z8 H(Clt(Z8);x) = 20+102x+88x2, W (Clt(Z8)) = 278, ad(Clt(Z8)) = 10.2, D(Clt(Z8)) = 1.46 Z10 (Clt(Z10);x) = 12 + 50x+ 16x2, W (Clt(Z10)) = 82, ad(Clt(Z10)) = 5, D(Clt(Z10)) = 1.242 Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1645 Ring Graph Z12 H(Clt(Z12);x) = 36 + 310x + 320x2, W (Clt(Z12)) = 950, ad(Clt(Z12)) = 17.222, D(Clt(Z12)) = 1.50 Z15 H(Clt(Z15);x) = 36 + 310x + 320x2 W (Clt(Z15)) = 950, ad(Clt(Z15)) = 17.222, D(Clt(Z15)) = 1.50 Z20 H(Clt(Z20);x) = 36 + 310x + 320x2, W (Clt(Z20)) = 950, ad(Clt(Z20)) = 17.222, D(Clt(Z20)) = 1.50 Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1646 Ring Graph Z24 H(Clt(Z24);x) = 120 + 2156x + 4984x2, W (Clt(Z24)) = 12124, ad(Clt(Z24)) = 35.9333, D(Clt(Z24)) = 1.69 Z30 H(Clt(Z30);x) = 72 + 1096x + 1460x2, W (Clt(Z30)) = 4016, ad(Clt(Z30)) = 30.444, D(Clt(Z30)) = 1.57 Z40 H(Clt(Z40);x) = 120 + 2156x + 4984x2, W (Clt(Z40)) = 12124, ad(Clt(Z40)) = 35.9333, D(Clt(Z40)) = 1.69 Mohammed Al-Neima et al. / Eur. J. Pure Appl. Math, 15 (4) (2022), 1637-1648 1647 Ring Graph Z60 H(Clt(Z60);x) = 216 + 6412x + 16808x2 W (Clt(Z60)) = 40028, ad(Clt(Z60)) = 59.370, D(Clt(Z60)) = 1.72 Z120 H(Clt(Z120);x) = 720+43192x+215648x2, W (Clt(Z120)) = 474488, ad(Clt(Z120)) = 119.97, D(Clt(Z60)) = 1.83 Theorem 3.2. The graph Clt(R) is connected for any invo-t-clean ring and has a diameter less than and equal 2. Proof. Since (u, 0) belong to V (Clt(R)), then (u, 0) is adjacent to all vertices in a graph Clt(R). Hence then Clt(R) is connected. Now to prove that Clt(R) has diameter less and equal 2. Let (u1, t1),(u2, t2) ∈ V (Clt(R)), then we have the following two cases: Case1: If t1 = 0, then d((u1, t1), (u2, t2)) = 1. Case2: If u1 + u2 ̸= 0 and t1 · t2 ̸= 0 , ui ∈ U2(R), ti ∈ Tri(R), i = 1, 2,then d((u1, t1), (u2, t2)) = 2. ■ Theorem 3.3. The graph Clt(R) has girth g(Clt(R)) equal three for all |Clt(R)| ≥ 3. Proof. Since R is a ring with identity, there are two vertices (1, 0) and (−1, 0) belong to V (Clt(R)). 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