� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 34, Num. 1 (2019), 61 – 76 doi:10.17398/2605-5686.34.1.61 Available online February 3, 2019 Additivity of elementary maps on gamma rings B.L. Macedo Ferreira Comission of Mathematicians, Federal Technological University of Paraná Professora Laura Pacheco Bastos Avenue, 800, 85053–510, Guarapuava, Brazil brunoferreira@utfpr.edu.br Received January 11, 2018 Presented by Alexandre Turull Accepted October 15, 2018 Abstract: Let M and M′ be Gamma rings, respectively. We study the additivity of surjective elementary maps of M × M′. We prove that if M contains a non-trivial γ-idempotent satisfying some conditions, then they are additive. Key words: Elementary maps, Gamma rings, additivity. AMS Subject Class. (2010): 16Y99. 1. Gamma rings and elementary maps Let M and Γ be two abelian groups. We call M a Γ-ring if the following conditions are satisfied: (i) xαy ∈ M, (ii) (x + y)αz = xαz + yαz, xα(y + z) = xαy + xαz, (iii) x(α + β)y = xαy + xβy, (iv) (xαy)βz = xα(yβz), for all x,y,z ∈ M and α,β ∈ Γ. N. Nobusawa introduced the notion of a Γ-ring, more general than a ring in his paper entitled “On a generalization of the ring theory”. For those readers who are not familiar with this language of Γ-rings we recommend “On a generalization of the ring theory” and “On the Γ-rings of Nobusawa” [2] and [1] respectively. Our purpose in this paper is the study of the additivity of a specific application on Γ-rings, for this we will address some preliminary definitions. A nonzero element 1 ∈ M is called a multiplicative γ-identity of M or γ-unity element (for some γ ∈ Γ) if 1γx = xγ1 = x for all x ∈ M. A nonzero element e1 ∈ M is called a γ1-idempotent (for some γ1 ∈ Γ) if e1γ1e1 = e1 and ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.61 mailto:brunoferreira@utfpr.edu.br https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 62 b.l.m. ferreira a nontrivial γ1-idempotent if it is a γ1-idempotent different from multiplicative γ1-identity element of M. Let Γ, Γ′, M and M′ be additive groups such that M is a Γ-ring and M′ is a Γ′-ring. Let M : M → M′ and M∗ : M′ → M be two maps and φ : Γ → Γ′, φ∗ : Γ′ → Γ two bijective maps. We call the ordered pair (M,M∗) an elementary map of M×M′ if M(aαM∗(x)βb) = M(a)φ(α)xφ(β)M(b), M∗(xµM(a)νy) = M∗(x)φ∗(µ)aφ∗(ν)M∗(y) for all α,β ∈ Γ, a,b ∈ M, µ,ν ∈ Γ′ and x,y ∈ M′. We say that the elementary map (M,M∗) of M × M′ is additive (resp., injective, surjective, bijective) if both maps M and M∗ are additive (resp., injective, surjective, bijective). Let M and Γ be two abelian groups such that M is a Γ-ring and e1 ∈ M a nontrivial γ1-idempotent. Let us consider e2 : Γ × M → M, e′2 : M × Γ → M two M-additive maps such that e2(γ1,a) = a− e1γ1a, e′2(a,γ1) = a−aγ1e1. Let us denote e2αa = e2(α,a), aαe2 = e ′ 2(a,α), 11αa = e1αa + e2αa, aα11 = aαe1 + aαe2 and suppose (aαe2)βb = aα(e2βb) for all α,β ∈ Γ and a,b ∈ M. Then 11γ1a = aγ111 = a and (aα11)βb = aα(11βb), for all α,β ∈ Γ and a,b ∈ M, and M has a Peirce decomposition M = M11 ⊕M12 ⊕M21 ⊕M22, where Mij = eiγ1Mγ1ej (i,j = 1, 2), satisfying the multiplicative relations: (i) MijΓMjl ⊆ Mil(i,j, l = 1, 2); (ii) Mijγ1Mkl = 0 if j 6= k (i,j,k, l = 1, 2). If A and B are subsets of a Γ-ring M and Θ ⊆ Γ, we denote AΘB the subset of M consisting of all finite sums of the form ∑ i aiγibi where ai ∈ A, γi ∈ Θ and bi ∈ B. A right ideal (resp., left ideal) of a Γ-ring M is an additive subgroup I of M such that IΓM ⊆ I (resp., MΓI ⊆ I). If I is both a right and a left ideal of M, then we say that I is an ideal or two-side ideal of M. An ideal P of a Γ-ring M is called prime if for any ideals A,B ⊆ M, AΓB ⊆ P implies that A ⊆ P or B ⊆ P. A Γ-ring M is said to be prime if the zero ideal is prime. Theorem 1.1. ([9, Theorem 4]) If M is a Γ-ring, the following condi- tions are equivalent: (i) M is a prime Γ-ring; (ii) if a,b ∈ M and aΓMΓb = 0, then a = 0 or b = 0. additivity of elementary maps on gamma rings 63 The first result about the additivity of maps on rings was given by Martin- dale III in an excellent paper [10]. He established a condition on a ring M such that every multiplicative bijective map on M is additive. Li and Lu [8] also considered this question in the context of prime associative rings containing a nontrivial idempotent. They proved the following theorem. Theorem 1.2. Let M and M′ be two associative rings. Suppose that M is a 2-torsion free ring containing a family {eα : α ∈ Λ} of idempotents which satisfies: (i) If x ∈ M is such that xM = 0, then x = 0; (ii) If x ∈ M is such that eαMx = 0 for all α ∈ Λ, then x = 0 (and hence Mx = 0 implies x = 0); (iii) For each α ∈ Λ and x ∈ M, if eαxeαM(1 −eα) = 0 then eαxeα = 0. Then every surjective elementary map (M,M∗) of M×M′ is additive. During the last decade, many mathematicians devoted to study the ad- ditivity of maps on associative rings. However, is very difficult to say any- thing when these applications are defined on arbitrary rings which are not necessarily associative. For the reader interested in applications defined in non-associative rings we recommend some papers [3, 4, 5, 6, 7]. Thus this motivated us in the present paper takes up the special case of an Γ-ring. We investigate the problem of when a elementary map must be an additive map on the class of Γ-rings. 2. The main result We will prove that every surjective elementary map (M,M∗) of M × M′ is additive for this we will assume that M contains a family {eα : α ∈ Λ} of γα-idempotents satisfying some conditions. Our main result reads as follows. Theorem 2.1. Let Γ, Γ′, M and M′ be additive groups such that M is a Γ-ring and M′ is a Γ′-ring. Suppose that M contains a family {eα : α ∈ Λ} of γα-idempotents which satisfies: (i) If x ∈ M is such that xΓM = 0, then x = 0; (ii) If x ∈ M is such that eαγαMΓx = 0 for all α ∈ Λ, then x = 0 (and hence MΓx = 0 implies x = 0); 64 b.l.m. ferreira (iii) For each α ∈ Λ and x ∈ M, if (eαγαxγαeα)ΓMΓ(1α − eα) = 0 then eαγαxγαeα = 0. Then every surjective elementary map (M,M∗) of M×M′ is additive. The following lemmas has the same hypotheses of Theorem 2.1 and we need these lemmas for the proof of this theorem. Thus, let us consider e1 ∈ {eα : α ∈ Λ} a nontrivial γ1-idempotent of M and 11 = e1 + e2. We begin with the following trivial lemma Lemma 2.1. M(0) = 0 and M∗(0) = 0. Proof. M(0) = M(0αM∗(0)β0) = M(0)φ(α)0φ(β)M(0) = 0. Similarly, we have M∗(0) = 0. Lemma 2.2. M and M∗ are bijective. Proof. It suffices to prove that M and M∗ are injective. First show that M is injective. Let x1 and x2 be in M and suppose that M(x1) = M(x2). Since M ∗(uµM(xi)νv) = M ∗(u)φ∗(µ)xiφ ∗(ν)M∗(v) (i = 1, 2) for all µ,ν ∈ Γ′ and u,v ∈ M′, it follows that M∗(u)φ∗(µ)x1φ∗(ν)M∗(v) = M∗(u)φ∗(µ)x2φ ∗(ν)M∗(v). Hence from the surjectivity of φ∗ and M∗ and conditions (i) and (ii) we conclude that x1 = x2. Now we turn to proving the injectivity of M∗. Let u1 and u2 be in M ′ and suppose M∗(u1) = M ∗(u2). Since M∗M(xαM∗(ui)βy) = M ∗(M(x)φ(α)uiφ(β)M(y)) = M∗ ( M(x)φ(α)MM−1(ui)φ(β)M(y) ) = M∗M(x)φ∗φ(α)M−1(ui)φ ∗φ(β)M∗M(y) for all α,β ∈ Γ and x,y ∈ M, it follows that M∗M(x)φ∗φ(α)M−1(u1)φ ∗φ(β)M∗M(y) = M∗M(x)φ∗φ(α)M−1(u2)φ ∗φ(β)M∗M(y). Noting that φ∗φ and M∗M are also surjective, we see that M−1(u1) = M−1(u2), by conditions (i) and (ii). Consequently u1 = u2. additivity of elementary maps on gamma rings 65 Lemma 2.3. The pair (M∗−1,M−1) is an elementary map of M × M′, that is, the maps M∗−1 : M → M′ and M−1 : M′ → M satisfy M∗ −1( aαM−1(x)βb ) = M∗ −1 (a)φ∗ −1 (α)xφ∗ −1 (β)M∗ −1 (b), M−1 ( xµM∗ −1 (a)νy ) = M−1(x)φ−1(µ)aφ−1(ν)M−1(y) for all α,β ∈ Γ, µ,ν ∈ Γ′, a,b ∈ M and x,y ∈ M′. Proof. The first equality can follow from M∗ ( M∗ −1 (a)φ∗ −1 (α)xφ∗ −1 (β)M∗ −1 (b) ) = M∗ ( M∗ −1 (a)φ∗ −1 (α)MM−1(x)φ∗ −1 (β)M∗ −1 (b) ) = aφ∗ ( φ∗ −1 (α) ) M−1(x)φ∗ ( φ∗ −1 (β) ) b = aαM−1(x)βb and the second equality follows in a similar way. Lemma 2.4. Let s,a,b ∈ M such that M(s) = M(a) + M(b). Then (i) M(sαxβy) = M(aαxβy) + M(bαxβy) for α,β ∈ Γ and x,y ∈ M; (ii) M(xαyβs) = M(xαyβa) + M(xαyβb) for α,β ∈ Γ and x,y ∈ M; (iii) M∗−1(xαsβy) = M∗−1(xαaβy) + M∗−1(xαbβy) for α,β ∈ Γ and x,y ∈ M for x,y ∈ M. Proof. (i) Let α,β ∈ Γ and x,y ∈ M. Then M(sαxβy) = M ( sαM∗M∗ −1 (x)βy ) = M(s)φ(α)M∗ −1 (x)φ(β)M(y) = ( M(a) + M(b) ) φ(α)M∗ −1 (x)φ(β)M(y) = M(a)φ(α)M∗ −1 (x)φ(β)M(y) + M(b)φ(α)M∗ −1 (x)φ(β)M(y) = M(aαxβy) + M(bαxβy). (ii) The proof is similar to (i). 66 b.l.m. ferreira (iii) Let x,y ∈ M. By Lemma 2.3 M∗ −1 (xαsβy) = M∗ −1 (xαM−1M(s)βy) = M∗ −1 (x)φ∗ −1 (α)M(s)φ∗ −1 (β)M∗ −1 (y) = M∗ −1 (x)φ∗ −1 (α) ( M(a) + M(b) ) φ∗ −1 (β)M∗ −1 (y) = M∗ −1 (x)φ∗ −1 (α)M(a)φ∗ −1 (β)M∗ −1 (y) + M∗ −1 (x)φ∗ −1 (α)M(b)φ∗ −1 (β)M∗ −1 (y) = M∗ −1 (xαaβy) + M∗ −1 (xαbβy). The proof is complete. Lemma 2.5. The following are true: (i) M(a11 + a12 + a21 + a22) = M(a11) + M(a12) + M(a21) + M(a22); (ii) M∗−1(a11 + a12 + a21 + a22) = M ∗−1(a11) + M ∗−1(a12) + M ∗−1(a21) + M∗−1(a22). Proof. By the surjectivity of M, there exists s ∈ M such that M(s) = M(a11) + M(a12) + M(a21) + M(a22). Now, for arbitrary α,β ∈ Γ, xi1 ∈ Mi1 and y1j ∈ M1j, we have M∗ −1 (xi1αe1γ1sγ1e1βy1j) = M∗ −1 (xi1αe1γ1a11γ1e1βy1j) + M ∗−1(xi1αe1γ1a12γ1e1βy1j) + M∗ −1 (xi1αe1γ1a21γ1e1βy1j) + M ∗−1(xi1αe1γ1a22γ1e1βy1j) = M∗ −1 (xi1αe1γ1a11γ1e1βy1j), which implies xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy1j = 0. (2.1) In a similar way, for y2j ∈ M2j we get that xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy2j = 0. (2.2) From (2.1) and (2.2) we conclude xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy = 0. additivity of elementary maps on gamma rings 67 In a similar way, for y1j ∈ M1j and y2j ∈ M2j we get that xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy1j = 0, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy2j = 0, respectively, which implies xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy = 0. Thus, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ111βy = 0, for all β ∈ Γ, y ∈ M, that is, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) ΓM = 0. By condition (i) of the Theorem we have xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0. Repeating the above arguments, for xi2 ∈ Mi2 we get that xi2αe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0 which implies xαe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0. Similarly, we obtain xαe2γ1 ( s− (a11 + a12 + a21 + a22) ) = 0, which yields xα ( s− (a11 + a12 + a21 + a22) ) = 0. Thus, MΓ ( s− (a11 + a12 + a21 + a22) ) = 0 which results in s = a11 + a12 + a21 + a22. The proof of (ii) is similar, since the pair (M∗−1; M−1) is also an elemen- tary map of M×M′. Lemma 2.6. The following are true: (i) M(a12 + b12αa22) = M(a12) + M(b12αa22); 68 b.l.m. ferreira (ii) M(a11 + a12αa21) = M(a11) + M(a12αa21); (iii) M(a21 + a22αb21) = M(a21) + M(a22αb21). Proof. (i) From Lemma 2.5-(i) and (ii) we have M(a12 + b12αa22) = M ( e1γ1(e1 + b12)γ1(a12 + e2αa22) ) = M(e1)φ(γ1)M ∗−1(e1 + b12)φ(γ1)M(a12 + e2αa22) = M(e1)φ(γ1) ( M∗ −1 (e1) + M ∗−1(b12) ) φ(γ1) ( M(a12) + M(e2αa22) ) = M(e1)φ(γ1)M ∗−1(e1)φ(γ1)M(a12) + M(e1)φ(γ1)M ∗−1(e1)φ(γ1)M(e2αa22) + M(e1)φ(γ1)M ∗−1(b12)φ(γ1)M(a12) + M(e1)φ(γ1)M ∗−1(b12)φ(γ1)M(e2αa22) = M(e1γ1e1γ1a12) + M(e1γ1e1γ1a22) + M(e1γ1b12γ1a12) + M(e1γ1b12γ1e2αa22) = M(a12) + M(b12αa22). Note that we use properties (i) and (ii) in M(a12 + e2αa22) = M(a12) + M(e2αa22) and M∗ −1 (e1 + b12) = M ∗−1(e1) + M ∗−1(b12), respectively. So (i) follows. Observing that a11 + a12αa21 = (a11 + a12αe2)γ1(e1 + a21)γ1e1, a21 + a22αa21 = (a21 + a22αe2)γ1(e1 + b21)γ1e1, then (ii) and (iii) can be proved similarly. Lemma 2.7. M(a21γ1a12 + a22γ1b22) = M(a21γ1a12) + M(a22γ1b22). Proof. We first claim that M(a21γ1a12γ1c22 + a22γ1b22γ1c22) = M(a21γ1a12γ1c22) + M(a22γ1b22γ1c22) (2.3) additivity of elementary maps on gamma rings 69 holds for all c22 ∈ M22. Indeed, from Lemma 2.5-(i) and (ii), we see that M(a21γ1a12αc22 + a22γ1b22αc22) = M ( (a21 + a22)γ1(a12 + b22)αc22 ) = M(a21 + a22)φ(γ1)M ∗−1(a12 + b22)φ(α)M(c22) = ( M(a21) + M(a22) ) φ(γ1) ( M∗ −1 (a12) + M ∗−1(b22) ) φ(α)M(c22) = M(a21)φ(γ1)M ∗−1(a12)φ(α)M(c22) + M(a21)φ(γ1)M ∗−1(b22)φ(α)M(c22) + M(a22)φ(γ1)M ∗−1(a12)φ(α)M(c22) + M(a22)φ(γ1)M ∗−1(b22)φ(α)M(c22) = M(a21γ1a12αc22) + M(a21γ1b22αc22) + M(a22γ1a12αc22) + M(a22γ1b22αc22) = M(a21γ1a12αc22) + M(a22γ1b22αc22), as desired. Now we find s ∈ M such that M(s) = M(a21γ1a12) +M(a22γ1b22). For arbitrary element x21 ∈ M21, by Lemma 2.4-(i) and Lemma 2.6-(iii), M(sαx21) = M(sαx21γ1e1) = M ( (a21γ1a12)αx21γ1e1 ) + M ( (a22γ1b22)αx21γ1e1 ) = M(a21γ1a12αx21 + a22γ1b22αx21). It follows that ( s− (a21γ1a12 + a22γ1b22) ) αx21 = 0, (2.4) for all α ∈ Γ. Our next step will be to prove that( s− (a21γ1a12 + a22γ1b22) ) αx22 = 0 (2.5) holds for every x22 ∈ M22. First, for y21, by Lemmas 2.4-(i) and Lemma 2.6-(iii) M(sαx22βy21) = M ( (a21γ1a12)αx22βy21 ) + M ( (a22γ1b22)αx22βy21 ) = M ( (a21γ1a12)αx22βy21 + (a22γ1b22)αx22βy21 ) , which implies that sαx22βy21 = (a21γ1a12)αx22βy21 + (a22γ1b22)αx22βy21. It follows that ( s− (a21γ1a12 + a22γ1b22) ) αx22 ·βy21 = 0. 70 b.l.m. ferreira For y22 ∈ M22, by Lemma 2.4-(i) and (2.3) we have M(sαx22βy22) = M ( (a21γ1a12)αx22βy22 ) + M ( (a22γ1b22)αx22βy22 ) = M ( (a21γ1a12)αx22βy22 ) + M ( (a22γ1b22)αx22βy22 ) = M ( a21γ1a12αx22βy22 ) + M ( (a22γ1b22)αx22βy22 ) = M ( a21γ1a12αx22βy22 + (a22γ1b22)αx22βy22 ) = M ( (a21γ1a12)αx22βy22 + (a22γ1b22)αx22βy22 ) yielding that sαx22βy22 = (a21γ1a12 + a22γ1b22)αx22βy22. It follows that( s− (a21γ1a12 + a22γ1b22) ) αx22βy22 = 0. Hence, we obtain that( s− (a21γ1a12 + a22γ1b22) ) αx22 · ΓM = 0. So Eq. (2.5) follows by Theorem 2.1 condition (i). From Eqs. (2.4) and (2.5), we can get that ( s−(a21γ1a12+a22γ1b22) ) ΓM = 0. Therefore, s = a21γ1a12 + a22γ1b22 by Theorem 2.1 condition (i) again. Taking Lemma 2.3 into account, still hold true when M is replaced by M∗−1, that is Lemma 2.8. The following are true: (i) M∗−1(a12 + b12αa22) = M ∗−1(a12) + M ∗−1(b12αa22). (ii) M∗−1(a11 + a12αa21) = M ∗−1(a11) + M ∗−1(a12αa21). (iii) M∗−1(a21 + a22αb21) = M ∗−1(a21) + M ∗−1(a22αb21). (iv) M∗−1(a21γ1a12 + a22γ1b22) = M ∗−1(a21γ1a12) + M ∗−1(a22γ1b22). Lemma 2.9. M(a12 + b12) = M(a12) + M(b12). Proof. Let s ∈ M such that M(s) = M(a12) + M(b12). For x1j ∈ M1j, applying Lemma 2.4-(i), M(sγ1e1αx1j) = M(a12γ1e1αx1j) + M(b12γ1e1αx1j) = 0. These equations show that sγ1e1αx1j = 0 = (a12 + b12)γ1e1αx1j. Hence,( s− (a12 + b12) ) γ1e1αx1j = 0. For all x2j ∈ M2j M∗ −1 (sγ1e1αx2j) = M ∗−1(a12γ1e1αx2j) + M ∗−1(b12γ1e1αx2j) = 0, additivity of elementary maps on gamma rings 71 which implies that ( s− (a12 + b12) ) γ1e1αx2j = 0. Thus ( s− (a12 + b12) ) γ1e1αx = 0, for all α ∈ Γ and x ∈ M, which implies( s− (a12 + b12) ) γ1e1ΓM = 0. For y11 ∈ M11, applying Lemma 2.4-(i),(ii), M(y11βe1γ1sγ1e2αx22) = M(y11βe1γ1a12γ1e2αx22) + M(y11βe1γ1b12γ1e2αx22) = M(y11βe1γ1a12γ1e2αx22) + M(y11βe1γ1b12γ1e2αx22) = M(y11βe1γ1(a12 + b12)γ1e2αx22) These equations show that y11βe1γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0. For all y21 ∈ M21 M∗ −1 (y21βe1γ1sγ1e2αy21) = M ∗−1(y21βe1γ1a12γ1e2αx22) + M∗ −1 (y21βe1γ1b12γ1e2αx22) = M∗ −1 (y21βe1γ1a12γ1e2αx22) + M∗ −1 (y21βe1γ1y21βe1γ1b12γ1e2αx22) = M∗ −1 (y21βe1γ1(a12 + b12)γ1e2αx22) = M∗ −1( y21βe1γ1(a12 + b12)γ1e2αx22 ) which implies that y21βe1γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0. For yi2 ∈ Mi2, applying Lemma 2.4-(i),(ii), M(yi2βe1γ1sγ1e2αx22) = M(yi2βe1γ1a12γ1e2αx22) + M(yi2βe1γ1b12γ1e2αx22) = 0, 72 b.l.m. ferreira which implies yi2βe1γ1sγ1e2αx22 = 0 = yi2βe1γ1(a12 + b12)γ1e2αx22, yβe1γ1 ( s−(a12 + b12) ) γ1e2αx22 = 0. For yij ∈ Mij, applying Lemma 2.4-(i),(ii), M(yijβe2γ1sγ1e2αx22) = M(yijβe2γ1a12γ1e2αx22) + M(yijβe2γ1b12γ1e2αx22) = 0, yijβe2γ1sγ1e2αx22 = 0 = yijβe2γ1(a12 + b12)γ1e2αx22. These equations show that yijβe2γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0, yβe2γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0, MΓ ( s− (a12 + b12) ) γ1e2αx22 = 0,( s− (a12 + b12) ) γ1e2αx22 = 0,( s− (a12 + b12) ) γ1e2αx = 0,( s− (a12 + b12) ) ΓM = 0, s = a12 + b12. Lemma 2.10. M(a11 + b11) = M(a11) + M(b11). Proof. Choose s = s11 + s12 + s21 + s22 ∈ M such that M(s) = M(a11) + M(b11). M(s) = M(e1γ1a11γ1e1) + M(e1γ1b11γ1e1) = M(e1)φ(γ1)M ∗−1(a11)φ(γ1)M(e1) + M(e1)φ(γ1)M ∗−1(b11)φ(γ1)M(e1) = M(e1)φ(γ1) ( M∗ −1 (a11) + M ∗−1(b11) ) φ(γ1)M(e1) = M(e1)φ(γ1) ( M∗ −1 (e1γ1a11γ1e1) + M ∗−1(e1γ1b11γ1e1) ) φ(γ1)M(e1) = M(e1)φ(γ1)M ∗−1(e1γ1sγ1e1)φ(γ1)M(e1) = M(e1γ1e1γ1sγ1e1γ1e1) = M(e1γ1sγ1e1) = M(s11). additivity of elementary maps on gamma rings 73 It follows that s = s11. Hence s− (a11 + b11) ∈ M11. First we let x11 ∈ M12 be arbitrary. Applying Lemma 2.4-(i) we get that M(sαe1γ1x11γ1e1βe2) = M(a11αe1γ1x11γ1e1βe2) + M(b11αe1γ1x11γ1e1βe2) = M(a11αe1γ1x11γ1e1βe2 + b11αe1γ1x11γ1e1βe2) = M ( (a11 + b11)αe1γ1x11γ1e1βe2 ) yielding that sαe1γ1x11γ1e1βe2 = (a11 + b11)αe1γ1x11γ1e1βe2. Therefore( s− (a11 + b11) ) αe1γ1x11γ1e1βe2 = 0. This implies ( s− (a11 + b11) ) αe1γ1xγ1e1βe2 = 0. (2.6) Second we let x12 ∈ M12 be arbitrary. Applying Lemma 2.4-(i) we get that M(sαe1γ1x12γ1e2βe2) = M(a11αe1γ1x12γ1e2βe2) + M(b11αe1γ1x12γ1e2βe2) = M(a11αe1γ1x12γ1e2βe2 + b11αe1γ1x12γ1e2βe2) = M ( (a11 + b11)αe1γ1x12γ1e2βe2 ) yielding that sαe1γ1x12γ1e2βe2 = (a11 + b11)αe1γ1x12γ1e2βe2. Therefore( s− (a11 + b11) ) αe1γ1x12γ1e2βe2 = 0. This implies ( s− (a11 + b11) ) αe1γ1xγ1e2βe2 = 0. (2.7) Third we let x21 ∈ M21 be arbitrary. Applying Lemma 2.4-(i) we get that M(sαe2γ1x21γ1e1βe2) = M(a11αe2γ1x21γ1e1βe2) + M(b11αe2γ1x21γ1e1βe2) = M(a11αe2γ1x21γ1e1βe2 + b11αe2γ1x21γ1e1βe2) = M ( (a11 + b11)αe2γ1x21γ1e1βe2 ) yielding that sαe2γ1x21γ1e1βe2 = (a11 + b11)αe2γ1x21γ1e1βe2. Therefore( s− (a11 + b11) ) αe2γ1x21γ1e1βe2 = 0. 74 b.l.m. ferreira This implies ( s− (a11 + b11) ) αe2γ1xγ1e1βe2 = 0. (2.8) Lastly we let x22 ∈ M22 be arbitrary. Applying Lemma 2.4-(i) we get that M(sαe2γ1x22γ1e2βe2) = M(a11αe2γ1x22γ1e2βe2) + M(b11αe2γ1x22γ1e2βe2) = M(a11αe2γ1x22γ1e2βe2 + b11αe2γ1x22γ1e2βe2) = M ( (a11 + b11)αe2γ1x22γ1e2βe2 ) yielding that sαe2γ1x22γ1e2βe2 = (a11 + b11)αe2γ1x22γ1e2βe2. Therefore( s− (a11 + b11) ) αe2γ1x22γ1e2βe2 = 0. This implies ( s− (a11 + b11) ) αe2γ1xγ1e2βe2 = 0. (2.9) From (2.6)-(2.9) we have( s− (a11 + b11) ) α11γ1xγ111βe2 = 0, which implies ( s− (a11 + b11) ) αxβe2 = 0, for all α,β ∈ Γ and x ∈ M, which yields( s− (a11 + b11) ) ΓMΓe2 = 0. So ( s− (a11 + b11) ) αMβ(11 −e1) = 0 which implies( e1γ1(s− (a11 + b11)γ1e1 ) ΓMΓ(11 −e1) = 0. It follows, from Theorem 2.1 condition (iii), that s = a11 + b11. Lemma 2.11. M is additive on e1γ1M = M11 + M12. Proof. The proof is the same as that of Martindale III (1969, Lemma 5) and is included for the sake of completeness. In fact, let a11,b11 ∈ M11 and a12,b12 ∈ M12. Making use of Lemmas 2.5, 2.9 and 2.10 we can see that M ( (a11 + a12) + (b11 + b12) ) = M ( (a11 + b11) + (a12 + b12) ) = M(a11 + b11) + M(a12 + b12) = M(a11) + M(b11) + M(a12) + M(b12) = M(a11 + a12) + M(b11 + b12). holds true, as desired. additivity of elementary maps on gamma rings 75 Proof of Theorem 2.1. Suppose that a,b ∈ M and choose s ∈ M such that M(s) = M(a) + M(b). For all α ∈ Λ, M is additive on eαγαM because of Lemma 2.11. Thus, for every r ∈ M, we have M(eαγαrµs) = M(eα)φ(γα)M ∗−1(r)φ(µ)M(s) = M(eα)φ(γα)M ∗−1(r)φ(µ) ( M(a) + M(b) ) = M(eα)φ(γα)M ∗−1(r)φ(µ)M(a) + M(eα)φ(γα)M ∗−1(r)φ(µ)M(b) = M(eαγαrµa) + M(eαγαrµb) = M(eαγαrµa + eαγαrµb) = M ( eαγαrµ(a + b) ) . So eαγαrµs = eαγαrµ(a + b). Therefore eαγαMΓ ( s− (a + b) ) = 0 holds for every α ∈ Λ. We then conclude that s = a + b from Theorem 2.1 condition (ii). This shows that M is additive on M. To prove the additivity of M∗, let x,y ∈ M′. For a,b ∈ M, by using the additivity of M, we have M ( aλ ( M∗(x) + M∗(y) ) µb ) = M ( aλM∗(x)µb ) + M ( aλM∗(y)µb ) = M(a)φ(λ)xφ(µ)M(b) + M(a)φ(λ)yφ(µ)M(b) = M(a)φ(λ)(x + y)φ(µ)M(b) = M ( aλM∗(x + y)µb ) . It follows that aλ ( M∗(x) + M∗(y)−M∗(x + y) ) µb = 0 holds for all a,b ∈ M, that is, aλ ( M∗(x) + M∗(y) −M∗(x + y) ) ΓM = 0 holds for all a ∈ M, which implies aλ ( M∗(x) + M∗(y) −M∗(x + y) ) = 0 holds for all a ∈ M, which implies MΓ ( M∗(x) + M∗(y) −M∗(x + y) ) = 0 which forces M∗(x + y) = M∗(x) + M∗(y) because of Theorem 2.1 conditions (i) and (ii). This completes the proof. 76 b.l.m. ferreira Corollary 2.1. Let Γ, Γ′, M and M′ be additive groups such that M is a Γ-ring and M′ is a Γ′-ring such that M is a prime Γ-ring containing a non-trivial γ-idempotent (M need not have an γ-identity element), where γ ∈ Γ. Suppose e2 : Γ×M → M, e′2 : M×Γ → M two M-additive maps such that e2(γ1,a) = a − e1γ1a, e′2(a,γ1) = a − aγ1e1. Denote e2αa = e2(α,a), aαe2 = e ′ 2(a,α), 11αa = e1αa + e2αa, aα11 = aαe1 + aαe2 and suppose (aαe2)βb = aα(e2βb) for all α,β ∈ Γ and a,b ∈ M. Then every surjective elementary map (M,M∗) of M×M′ is additive. Proof. The result follows directly from Theorem 2.1 and Theorem 1.1. Corollary 2.2. Let Γ, Γ′, M and M′ be additive groups such that M is a Γ-ring and M′ is a Γ′-ring such that M is a prime Γ-ring containing a non-trivial γ-idempotent and a γ-unity element, where γ ∈ Γ. Then every surjective elementary map (M,M∗) of M×M′ is additive. References [1] W.E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math. 18 (1966), 411 – 422. [2] N. Nobusawa, On a generalization of the ring theory, Osaka J. Math. 1 (1964), 81 – 89. [3] J.C.M. Ferreira, B.L.M. Ferreira, Additivity of n-multiplicative maps on alternative rings, Comm. Algebra 44 (2016), 1557 – 1568. [4] B.L.M. Ferreira, J.C.M. Ferreira, H. Guzzo, Jr., Jordan triple maps of alternative algebras, JP J. Algebra Number Theory Appl. 33 (2014), 25 – 33. [5] B.L.M. Ferreira, H. Guzzo Jr., J.C.M. Ferreira, Additivity of Jor- dan elementary maps on standard rings, Algebra Discrete Math. 18 (2014), 203 – 233. [6] B.L.M. Ferreira, R. Nascimento, Derivable maps on alternative rings, Recen 16 (2014), 9 – 15. [7] B.L.M. Ferreira, J.C.M. Ferreira, H. Guzzo Jr., Jordan triple elementary maps on alternative rings, Extracta Math. 29 (2014), 1 – 18. [8] P. Li, F. Lu, Additivity of elementary maps on rings, Comm. Algebra 32 (2004), 3725 – 3737. [9] S. Kyuno, On prime gamma rings, Pacific J. Math. 75 (1978), 185 – 190. [10] W.S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695 – 698. Gamma rings and elementary maps The main result