Additivity of elementary maps on alternative rings
Let ℜ and ℜ′ be alternative rings. In this article we investigate the additivity of surjective elementary maps of ℜ × ℜ′. As a main theorem, we prove that if ℜ idempotent satisfying some conditions, these maps are additive.
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irk-123456789-1884792023-03-03T01:26:53Z Additivity of elementary maps on alternative rings Ferreira, B.L.M. Let ℜ and ℜ′ be alternative rings. In this article we investigate the additivity of surjective elementary maps of ℜ × ℜ′. As a main theorem, we prove that if ℜ idempotent satisfying some conditions, these maps are additive. 2019 Article Additivity of elementary maps on alternative rings / B.L.M. Ferreira // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 94–106. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC: 17A36, 17D99 http://dspace.nbuv.gov.ua/handle/123456789/188479 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let ℜ and ℜ′ be alternative rings. In this article we investigate the additivity of surjective elementary maps of ℜ × ℜ′. As a main theorem, we prove that if ℜ idempotent satisfying some conditions, these maps are additive. |
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Ferreira, B.L.M. Additivity of elementary maps on alternative rings Algebra and Discrete Mathematics |
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Ferreira, B.L.M. |
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Additivity of elementary maps on alternative rings |
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Additivity of elementary maps on alternative rings |
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Additivity of elementary maps on alternative rings |
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Additivity of elementary maps on alternative rings |
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Additivity of elementary maps on alternative rings |
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additivity of elementary maps on alternative rings |
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Інститут прикладної математики і механіки НАН України |
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Additivity of elementary maps on alternative rings / B.L.M. Ferreira // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 1. — С. 94–106. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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“adm-n3” — 2019/10/20 — 9:35 — page 94 — #96
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 28 (2019). Number 1, pp. 94–106
c© Journal “Algebra and Discrete Mathematics”
Additivity of elementary maps
on alternative rings
B. L. M. Ferreira
Communicated by I. P. Shestakov
Abstract. Let R and R
′ be alternative rings. In this arti-
cle we investigate the additivity of surjective elementary maps of
R×R
′. As a main theorem, we prove that if R contains a non-trivial
idempotent satisfying some conditions, these maps are additive.
1. Alternative rings and elementary maps
Let R be a ring not necessarily associative or commutative and consider
the following convention for its multiplication operation: xy · z = (xy)z
and x · yz = x(yz) for x, y, z ∈ R, in order to reduce the number of
parentheses. We denote the associator of R by (x, y, z) = xy · z − x · yz
for x, y, z ∈ R.
Let X = {xi}i∈N be an arbitrary set of variables. A non-associative
monomial of degree 1 is any element of X. Given a natural number n > 1,
a non-associative monomial of degree n is an expression of the form (u)(v),
where u is a non-associative monomial of some degree i and v is a non-
associative monomial of degree n− i. A non-associative polynomial f over
a ring R is any formal linear combination of non-associative monomials
with coefficients in R. If f includes no variables except x1, x2, . . . , xn and
a1, a2, . . . , an is a set of elements of R, then f(a1, a2, . . . , an) is an element
of R which results by applying the sequence of operations forming f to
a1, a2, . . . , an in place of x1, x2, . . . , xn.
2010 MSC: 17A36, 17D99.
Key words and phrases: elementary maps, alternative rings, additivity.
“adm-n3” — 2019/10/20 — 9:35 — page 95 — #97
B. L. M. Ferreira 95
Let R and R
′ be two rings and let M : R → R
′ and M∗ : R′ → R
be two maps. We call the ordered pair (M,M∗) an elementary map of
R×R
′ if for all non-associative monomial f = f(x1, x2, x3) of degree 3
M
(
f
(
a,M∗(x), b
))
= f
(
M(a), x,M(b)
)
,
M∗
(
f
(
x,M(a), y
))
= f
(
M∗(x), a,M∗(y)
)
for all a, b ∈ R and x, y ∈ R
′.
We say that the elementary map (M,M∗) of R×R
′ is additive (resp.,
injective, surjective, bijective) if both maps M and M∗ are additive (resp.,
injective, surjective, bijective).
A ring R is said to be alternative if (x, x, y) = 0 = (y, x, x) for all
x, y ∈ R. It is easily seen that any associative ring is an alternative ring.
An alternative ring R is called k-torsion free if k x = 0 implies x = 0,
for any x ∈ R, where k ∈ Z, k > 0, and prime if AB 6= 0 for any two
nonzero ideals A,B ⊆ R.
Let us consider R an alternative ring and fix a non-trivial idempotent
e1 ∈ R, i.e, e2
1
= e1, e1 6= 0 and e1 is not a unity element. Let e2 : R → R
and e′
2
: R → R be linear operators given by e2(a) = a− e1a and e′
2
(a) =
a−ae1. Clearly e2
2
= e2, (e
′
2
)2 = e′
2
and we note that if R has a unity, then
we can consider e2 = 1 − e1 ∈ R. Let us denote e2(a) by e2a and e′
2
(a)
by ae2. It is easy to see that eia · ej = ei · aej (i, j = 1, 2) for all a ∈ R.
Then R has a Peirce decomposition R = R11 ⊕R12 ⊕R21 ⊕R22, where
Rij = eiRej (i, j = 1, 2), (see [3]) satisfying the multiplicative relations:
(i) RijRjl ⊆ Ril (i, j, l = 1, 2);
(ii) RijRij ⊆ Rji (i, j = 1, 2);
(iii) RijRkl = 0, if j 6= k and (i, j) 6= (k, l), (i, j, k, l = 1, 2);
(iv) x2ij = 0, for all xij ∈ Rij (i, j = 1, 2; i 6= j).
According to [4], “The first surprising result on how the multiplicative
structure of a ring determines its additive structure is due to Martindale III
(1969). In Martindale III (1969, [5] Theorem), he established a condition
on a ring R such that every multiplicative bijective map on R is additive."
Li and Lu [4] also considered this question in the context of associative
rings containing a non-trivial idempotent. They proved the following
theorem.
Theorem 1. [4, Li and Lu] Let R and R
′ be two associative rings. Sup-
pose that R is a 2-torsion free ring containing a family {eα|α ∈ Λ} of
idempotents which satisfies:
(i) If x ∈ R is such that xR = 0, then x = 0;
“adm-n3” — 2019/10/20 — 9:35 — page 96 — #98
96 Additivity of elementary maps on alternative rings
(ii) If x ∈ R is such that eαRx = 0 for all α ∈ Λ, then x = 0 (and
hence Rx = 0 implies x = 0);
(iii) For each α ∈ Λ and x ∈ R, if eαxeαR(1− eα) = 0 then eαxeα = 0.
Then every surjective elementary map (M,M∗) of R×R
′ is additive.
The hypotheses in Li and Lu’s Theorem [4] allowed the author to
make its proof based on calculus using Peirce decomposition notion for
associative rings.
The notion of Peirce decomposition for alternative rings is similar
to that one for associative rings. However, this similarity is restricted
to its written form, not including its theoretical structure since Peirce
decomposition for alternative rings is a generalization of that classical
one for associative rings. Taking this fact into account, in the present
paper we generalize the main Li and Lu’s Theorem [4] to the class of
alternative rings. For this purpose, we adopt and follow the same structure
of the proof proposed by [4], in order to preserve the author’s ideas and
to highlight the generalization of associative results to the alternative
results. Therefore, our lemmas and the main theorem, which seem to
be equal in written form to those presented in lemmas and the theorem
proposed in Li and Lu [4], are distinguished by a fundamental item: the
use of the non-associative multiplications. The symbol “ ·”, as defined
in the introduction section of this article, is essential to elucidate how
the non-associative multiplication should be done, and also the symbol
“·” is used to simplify the notation. Therefore, the symbol “ ·” is crucial
to the logic, characterization and generalization of associative results to
alternative results.
2. The main result
Our main result reads as follows.
Theorem 2. Let R and R
′ be two alternative rings. Suppose that R is a
2-torsion free ring containing a family {eα|α ∈ Λ} of idempotents which
satisfies:
(i) If x ∈ R is such that xR = 0, then x = 0;
(ii) If x ∈ R is such that eαR · x = 0 (or eα ·Rx = 0) for all α ∈ Λ,
then x = 0 (and hence Rx = 0 implies x = 0);
(iii) For each α ∈ Λ and x ∈ R, if (eαxeα)·R(1−eα) = 0 then eαxeα = 0.
Then every surjective elementary map (M,M∗) of R×R
′ is additive.
“adm-n3” — 2019/10/20 — 9:35 — page 97 — #99
B. L. M. Ferreira 97
For proving Theorem 2 some preparatory material is needed, following
same steps as [4]. Firstly, we consider the case when the monomial is of
type f = f(x1, x2, x3). We begin with the following lemma.
Lemma 1. M(0) = 0 and M∗(0) = 0.
Proof. M(0) = M
(
0M∗(0) · 0
)
= M(0)0 ·M(0) = 0. Similarly, we have
M∗(0) = 0.
Lemma 2. M and M∗ are bijective.
Proof. It suffices to prove that M and M∗ are injective. We first show that
M is injective. Let x1 and x2 be in R and suppose that M(x1) = M(x2).
Since M∗
(
uM(xi) · v
)
= M∗(u)xi · M
∗(v) (i = 1, 2) for all u, v ∈ R
′,
it follows that M∗(u)x1 · M∗(v) = M∗(u)x2 · M∗(v). Hence from the
surjectivity of 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 R
′ and
suppose M∗(u1) = M∗(u2). Since
M∗M
(
xM∗(ui) · y
)
= M∗
(
M(x)ui ·M(y)
)
= M∗
(
M(x)MM−1(ui) ·M(y)
)
= M∗M(x)M−1(ui) ·M
∗M(y)
for all x, y ∈ R, it follows that
M∗M(x)M−1(u1) ·M
∗M(y) = M∗M(x)M−1(u2) ·M
∗M(y).
Noting that M∗M is also surjective, we see that M−1(u1) = M−1(u2), by
conditions (i) and (ii). Consequently u1 = u2.
Lemma 3. The pair (M∗−1,M−1) is an elementary map of R×R
′, that
is, the maps M∗−1 : R → R
′ and M−1 : R′ → R satisfy
M∗−1
(
aM−1(x) · b
)
= M∗−1(a)x ·M∗−1(b),
M−1
(
xM∗−1(a) · y
)
= M−1(x)a ·M−1(y)
for all a, b ∈ R and x, y ∈ R
′.
Proof. The first equality can be obtained from
M∗
(
M∗−1(a)x ·M∗−1(b)
)
= M∗
(
M∗−1(a)MM−1(x) ·M∗−1(b)
)
= aM−1(x) · b
and the second one follows in a similar way.
“adm-n3” — 2019/10/20 — 9:35 — page 98 — #100
98 Additivity of elementary maps on alternative rings
Lemma 4. Let s, a, b ∈ R such that M(s) = M(a) +M(b). Then
(i) M(sx · y) = M(ax · y) +M(bx · y) for x, y ∈ R;
(ii) M(xy · s) = M(xy · a) +M(xy · b) for x, y ∈ R;
(iii) M∗−1(xs·y) = M∗−1(xa·y)+M∗−1(xb·y) for x, y ∈ R for x, y ∈ R;
(iv) M(s · xy) = M(a · xy) +M(b · xy) for x, y ∈ R;
(v) M(x · ys) = M(x · ya) +M(x · yb) for x, y ∈ R;
(vi) M∗−1(x·sy) = M∗−1(x·ay)+M∗−1(x·by) for x, y ∈ R for x, y ∈ R.
Proof. (i) Let x, y ∈ R. Then
M(sx · y) = M
(
sM∗M∗−1(x) · y
)
= M(s)M∗−1(x) ·M(y)
=
(
M(a) +M(b)
)
M∗−1(x) ·M(y)
= M(a)M∗−1(x) ·M(y) +M(b)M∗−1(x) ·M(y)
= M(ax · y) +M(bx · y).
(ii) Let x, y ∈ R. Then
M(xy · s) = M
(
xM∗M∗−1(y) · s
)
= M(x)M∗−1(y) ·M(s)
= M(x)M∗−1(y) ·
(
M(a) +M(b)
)
= M(x)M∗−1(y) ·M(a) +M(x)M∗−1(y) ·M(b)
= M(xy · a) +M(xy · b).
(iii) Let x, y ∈ R. By Lemma 2.3
M∗−1(xs · y) = M∗−1(xM−1M(s) · y)
= M∗−1(x)M(s) ·M∗−1(y)
= M∗−1(x)
(
M(a) +M(b)
)
·M∗−1(y)
= M∗−1(x)M(a) ·M∗−1(y) +M∗−1(x)M(b) ·M∗−1(y)
= M∗−1(xa · y) +M∗−1(xb · y).
Similarly, we prove (iv), (v) and (vi), which finishes the proof.
Lemma 5. The following statements are true:
(i) M(a11 + a12) = M(a11) +M(a12);
(ii) M∗−1(a11 + a12) = M∗−1(a11) +M∗−1(a12).
“adm-n3” — 2019/10/20 — 9:35 — page 99 — #101
B. L. M. Ferreira 99
Proof. By surjectivity of M , there exists s ∈ R such that M(s) = M(a11)+
M(a12). Now,
M(e1e1 · s) = M(e1e1 · a11) +M(e1e1 · a12) = M(s).
It follows that e1e1 · s = s which implies s21 = s22 = 0. Also
M(s · e1e1) = M(a11 · e1e1) +M(a12 · e1e1) = M(a11).
From this equality we get s · e1e1 = a11 and therefore s11 = a11.
For an arbitrary b12 ∈ R12, we obtain
M(sb12 · e1) = M(a11b12 · e1) +M(a12b12 · e1) = M(a12b12 · e1),
which implies sb12 · e1 = a12b12 · e1, or still (s12 − a12)b12 = 0. In a similar
way, for an arbitrary b21 ∈ R21, we have
M(sb21 · e1) = M(a11b21 · e1) +M(a12b21 · e1) = M(a12b21 · e1).
Hence sb21 · e1 = a12b21 · e1 and thus (s12 − a12)b21 = 0. Finally, for
b22 ∈ R22,
M∗−1(e1s · b22) = M∗−1(e1a11 · b22) +M∗−1(e1a12 · b22)
= M∗−1(e1a12 · b22).
As a consequence, e1s · b22 = e1a12 · b22 which implies (s12 − a12)b22 = 0.
From these considerations, (s12 − a12)R = 0. According to (i), s12 = a12.
Similarly, we prove the lemma below.
Lemma 6. The following statements are true:
(i) M(a11 + a21) = M(a11) +M(a21);
(ii) M∗−1(a11 + a21) = M∗−1(a11) +M∗−1(a21).
Lemma 7. The following statements 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 surjectivity of M , there exists s ∈ R such that M(s) = M(a11)+
M(a12) +M(a21) +M(a22). Now,
M(e1e1 · s) = M(e1e1 · a11) +M(e1e1 · a12) = M(a11 + a12).
“adm-n3” — 2019/10/20 — 9:35 — page 100 — #102
100 Additivity of elementary maps on alternative rings
It follows from this equality that e1e1 ·s = a11+a12 which implies s11 = a11
and s12 = s12. Also
M(s · e1e1) = M(a11 · e1e1) +M(a21 · e1e1) = M(a11 + a21),
from where we get s · e1e1 = a11 + a21, or still s21 = a21.
For arbitrary b22, c22 ∈ R22, we have
M∗−1(b22s · c22) = M∗−1(b22a11 · c22) +M∗−1(b22a12 · c22)
+M∗−1(b22a21 · c22) +M∗−1(b22a22 · c22)
= M∗−1(b22a22 · c22).
Hence b22s · c22 = b22a22 · c22 which implies b22(s22 − a22) · c22 = 0. Now,
for an arbitrary c21 ∈ R21, we have
M∗−1(b22s · c21) = M∗−1(b22a11 · c21) +M∗−1(b22a12 · c21)
+M∗−1(b22a21 · c21) +M∗−1(b22a22 · c21)
= M∗−1(b22a21 · c21 + b22a22 · c21).
Thus b22s·c21 = b22a21 ·c21+b22a22 ·c21 and therefore b22(s22−a22)·c21 = 0.
As a consequence it follows that b22(s22 − a22) · R = 0 which implies
b22(s22 − a22) = 0. Finally, for an arbitrary b12 ∈ R12, we have
M(e1 · b12s) = M(e1 · b12a11) +M(e1 · b12a12)
+M(e1 · b12a21) +M(e1 · b12a22)
= M(e1 · b12a21 + e1 · b12a22).
Hence e1 · b12s = e1 · b12a21 + e1 · b12a22 which implies b12(s22 − a22) = 0.
Consequently R(s22 − a22) = 0. By condition (i), we have s22 = a22.
The proof of (ii) is similar, since the pair (M∗−1;M−1) is also an
elementary map of R×R
′.
Lemma 8. The following statements are true:
(i) M(a12 + b21c21) = M(a12) +M(b21c21);
(ii) M(a12 + b12a22) = M(a12) +M(b12a22);
(iii) M(a11 + a12a21) = M(a11) +M(a12a21);
(iv) M(a21 + a22b21) = M(a21) +M(a22b21).
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B. L. M. Ferreira 101
Proof. (i) Observing that a12 + b21c21 = e1 · (e1 + b21)(a12 + c21), we get
M(a12 + b21c21) = M
(
e1 · (e1 + b21)(a12 + c21)
)
= M(e1) ·M
∗−1(e1 + b21)M(a12 + c21)
= M(e1) ·
(
M∗−1(e1) +M∗−1(b21)
)(
M(a12) +M(c21)
)
= M(e1) ·M
∗−1(e1)M(a12) +M(e1) ·M
∗−1(e1)M(c21)
+M(e1) ·M
∗−1(b21)M(a12) +M(e1) ·M
∗−1(b21)M(c21)
= M(e1 · e1a12) +M(e1 · e1c21) +M(e1 · b21a12) +M(e1 · b21a21)
= M(a12) +M(b21a21).
(ii) From Lemma 7(i) and (ii) we have
M(a12 + b12a22) = M
(
e1 · (e1 + b12)(a12 + a22)
)
= M(e1) ·M
∗−1(e1 + b12)M(a12 + a22)
= M(e1) ·
(
M∗−1(e1) +M∗−1(b12)
)(
M(a12) +M(a22)
)
= M(e1) ·M
∗−1(e1)M(a12) +M(e1) ·M
∗−1(e1)M(a22)
+M(e1) ·M
∗−1(b12)M(a12) +M(e1) ·M
∗−1(b12)M(a22)
= M(e1 · e1a12) +M(e1 · e1a22) +M(e1 · b12a12) +M(e1 · b12a22)
= M(a12) +M(b12a22).
So (ii) follows. Observing that a11 + a12a21 = (a11 + a12)(e1 + a21) · e1
and a21 + a22b21 = (a21 + a22)(e1 + b21) · e1, then (iii) and (iv) can be
proved similarly.
Lemma 9. M(a21a12 + a22b22) = M(a21a12) +M(a22b22).
Proof. We first claim that M(a21a12 · c22+a22b22 · c22) = M(a21a12 · c22)+
M(a22b22 · c22) holds for all c22 ∈ R22. Indeed, from Lemma 7(i) and (ii),
we obtain
M(a21a12 · c22 + a22b22 · c22) = M
(
(a21 + a22)(a12 + b22) · c22
)
= M(a21 + a22)M
∗−1(a12 + b22) ·M(c22)
=
(
M(a21) +M(a22)
)(
M∗−1(a12) +M∗−1(b22)
)
·M(c22)
= M(a21)M
∗−1(a12) ·M(c22) +M(a21)M
∗−1(b22) ·M(c22)
+M(a22)M
∗−1(a12) ·M(c22) +M(a22)M
∗−1(b22) ·M(c22)
= M(a21a12 · c22) +M(a21b22 · c22) +M(a22a12 · c22)
+M(a22b22 · c22)
= M(a21a12 · c22) +M(a22b22 · c22),
“adm-n3” — 2019/10/20 — 9:35 — page 102 — #104
102 Additivity of elementary maps on alternative rings
as desired. Now let s ∈ R such that M(s) = M(a21a12) + M(a22b22),
which existence is ensured by surjectivity. Then
M(e1e1 · s) = M
(
e1e1 · (a21a12)
)
+M
(
e1e1 · (a22b22)
)
= 0.
Hence e1e1 · s = 0, or still s11 = s12 = 0. Similarly, we prove s21 = 0.
For an arbitrary element x21 ∈ R21, it follows from Lemma 4-(iv) that
M(sx21) = M(s · x21e1) = M
(
(a21a12) · x21e1
)
+M
(
(a22b22) · x21e1
)
= M(a21a12 · x21 + a22b22 · x21),
from where we get
(
s− (a21a12 + a22b22)
)
x21 = 0. (1)
As a next step we prove that
(
s− (a21a12 + a22b22)
)
x22 = 0 (2)
holds for every x22 ∈ R22. First, for y21, by Lemma 4-(i)
M(sx22 · y21) = M
(
(a21a12)x22 · y21
)
+M
(
(a22b22)x22 · y21
)
= M
(
(a21a12)x22 · y21 + (a22b22)x22 · y21
)
,
which implies that sx22 ·y21 = (a21a12)x22 ·y21+(a22b22)x22 ·y21. Therefore
(
s− (a21a12 + a22b22)
)
x22 · y21 = 0.
Similarly, for y22 ∈ R22, using Lemma 4(i)
M(sx22 · y22) = M
(
(a21a12)x22 · y22
)
+M
(
(a22b22)x22 · y22
)
= M
(
(a21a12)x22 · y22
)
+M
(
(a22b22)x22 · y22
)
= M
(
a21(a12x22) · y22
)
+M
(
(a22b22)x22 · y22
)
= M
(
a21(a12x22) · y22 + (a22b22)x22 · y22
)
= M
(
(a21a12)x22 · y22 + (a22b22)x22 · y22
)
yielding that sx22 · y22 = (a21a12 + a22b22)x22 · y22. Thus
(
s − (a21a12 +
a22b22)
)
x22 · y22 = 0 and therefore we obtain that
(
s− (a21a12 + a22b22)
)
x22 ·R = 0. (3)
Equation 3 follows from Theorem 2(i).
From Equation 1 and 2, we get that
(
s − (a21a12 + a22b22)
)
R = 0.
Hence s = a21a12 + a22b22 due to Theorem 2(i).
“adm-n3” — 2019/10/20 — 9:35 — page 103 — #105
B. L. M. Ferreira 103
Taking Lemma 3 into account, we point out that Lemma 9 can still
be obtained when M is replaced by M∗−1, as states the following lemma.
Lemma 10. The following are true:
(i) M∗−1(a12 + b12a22) = M∗−1(a12) +M∗−1(b12a22).
(ii) M∗−1(a11 + a12a21) = M∗−1(a11) +M∗−1(a12a21).
(iii) M∗−1(a21 + a22b21) = M∗−1(a21) +M∗−1(a22b21).
(iv) M∗−1(a21a12 + a22b22) = M∗−1(a21a12) +M∗−1(a22b22).
Lemma 11. M(a12 + b12) = M(a12) +M(b12).
Proof. Let s ∈ R be such that M(s) = M(a12) +M(b12). Then M(e1e1 ·
s) = M(e1e1 · a12) + M(e1e1 · b12) = M(s) and M(s · e1e1) = M(a12 ·
e1e1) + M(b12 · e1e1) = 0 which implies e1e1 · s = s and s · e1e1 = 0,
respectively. It follows that s21 + s22 = 0 and s11 + s21 = 0, respectively.
Thus s11 = s21 = s22 = 0.
For x21 ∈ R21, applying Lemma 4-(iv),
M(sx21) = M(s · x21e1) = M(a12 · x21e1) +M(b12 · x21e1)
= M(a12x21) +M(b12x21) = M(a12x21 + b12x21).
These above equations show that sx21 = (a12 + b12)x21. Hence
(
s− (a12 + b12)
)
x21 = 0. (4)
For all x22 ∈ R22
M∗−1(sx22) = M∗−1(e1 · sx22) = M∗−1(e1 · a12x22) +M∗−1(e1 · b12x22)
= M∗−1(a12x22) +M∗−1(b12x22) = M∗−1(a12x22 + b12x22)
which implies that
(
s− (a12 + b12)
)
x22 = 0. (5)
We now want to prove that
(
s − (a12 + b12)
)
x12 = 0, for all x12 ∈ R12.
Indeed, by Lemma 4-(vi), for y12 ∈ R12
M∗−1(y12 · sx12) = M∗−1(y12 · a12x12) +M∗−1(y12 · b12x12)
= M∗−1(y12 · a12x12 + y12 · b12x12).
We then get that y12 · sx12 = y12 · (a12x12 + b12x12) which implies y12 ·
(
s− (a12 + b12)
)
x12 = 0.
“adm-n3” — 2019/10/20 — 9:35 — page 104 — #106
104 Additivity of elementary maps on alternative rings
For y21 ∈ R21, from Lemma 4-(vi)
M∗−1(y21 · sx12) = M∗−1(y21 · a12x12) +M∗−1(y21 · b12x12)
= M∗−1(y21 · a12x12 + y21 · b12x12).
As a consequence y21 · sx12 = y21 · a12x12 + y21 · b12x12 which implies
y21 ·
(
s− (a12 + b12)
)
x12 = 0.
Now, for y22 ∈ R22, from Lemma 4-(vi)
M∗−1(y22 · sx12) = M∗−1(y22 · a12x12) +M∗−1(y22 · b12x12)
= M∗−1(y22 · a12x12 + y22 · b12x12).
From this, y22 · sx12 = y22 · a12x12 + y22 · b12x12 which implies y22 ·
(
s−
(a12 + b12)
)
x12 = 0.
Hence R ·
(
s− (a12 + b12)
)
x12 = 0, and therefore
(
s− (a12 + b12)
)
x12 = 0, (6)
according to Theorem 2(ii). Moreover, from Equations 4, 5 and 6, we get
that
(
s−(a12+b12)
)
R = 0. Due to Theorem 2(i) we have s = a12+b12.
Lemma 12. M(a11 + b11) = M(a11) +M(b11).
Proof. Choose s = s11 + s12 + s21 + s22 ∈ R such that M(s) = M(a11) +
M(b11). Using Lemma 4-(ii) and (iv) we have M(e1e1 · s) = M(s) and
M(s · e1e1) = M(s), which implies s21 + s22 = 0 and s12 + s22 = 0,
respectively. Hence s12 = s21 = s22 = 0.
These equations show that s = s11 and so s− (a11 + b11) ∈ R11. Next
let x12 ∈ R12 arbitrarily chosen. Applying Lemma 4-(iv) we get that
M(sx12) = M(s · e1x12) = M(a11 · e1x12) +M(b11 · e1x12)
= M(a11 · e1x12 + b11 · e1x12) = M(a11x12 + b11x12)
which yields sx12 = (a11 + b11)x12. Therefore
(
s − (a11 + b11)
)
R12 = 0
or still
(
s − (a11 + b11)
)
·R(1 − e1) = 0. Since s − (a11 + b11) ∈ R11, it
follows from Theorem 2(iii) that s = a11 + b11.
Lemma 13. M is additive on e1R = R11 +R12.
Proof. The proof is the same as that of Martindale III (1969, Lemma 5)
and is included for the sake of completeness. Indeed, let a11, b11 ∈ R11
“adm-n3” — 2019/10/20 — 9:35 — page 105 — #107
B. L. M. Ferreira 105
and a12, b12 ∈ R12. According to Lemmas 7, 11 and 12
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),
as desired.
Proof of Theorem 2. Suppose that a, b ∈ R and choose s ∈ R such that
M(s) = M(a) + M(b). For all α ∈ Λ, M is additive on eαR due to
Lemma 13. Thus, for every r ∈ R, we have
M(eα · rs) = M(eα) ·M
∗−1(r)M(s) = M(eα) ·M
∗−1(r)
(
M(a) +M(b)
)
= M(eα) ·M
∗−1(r)M(a) +M(eα) ·M
∗−1(r)M(b)
= M(eα · ra) +M(eα · rb) = M(eα · ra+ eα · rb)
= M
(
eα · r(a+ b)
)
,
and therefore eα · rs = eα · r(a + b). Hence eα ·R
(
s − (a + b)
)
= 0, for
every α ∈ Λ. We then conclude that s = a+ b from Theorem 2(ii). This
shows that M is additive on R.
In order to prove the additivity of M∗, let x, y ∈ R
′. For a, b ∈ R, by
using the additivity of M , we have
M
(
a
(
M∗(x) +M∗(y)
)
· b
)
= M
(
aM∗(x) · b
)
+M
(
aM∗(y) · b
)
= M(a)x ·M(b) +M(a)y ·M(b)
= M(a)(x+ y) ·M(b)
= M
(
aM∗(x+ y) · b
)
.
It follows that a
(
M∗(x)+M∗(y)−M∗(x+y)
)
·b = 0, for all a, b ∈ R, which
forces M∗(x+ y) = M∗(x) +M∗(y) according to Theorem 2, completing
the proof.
Corollary 1. Let R be a 2 and 3-torsion free prime alternative ring con-
taining a non-trivial idempotent (R need not have an identity element), and
let R′ be an arbitrary alternative ring. Then every surjective elementary
map (M,M∗) of R×R
′ is additive.
Proof. The result follows directly from Theorem 2 and [2, Theorem 2.2].
“adm-n3” — 2019/10/20 — 9:35 — page 106 — #108
106 Additivity of elementary maps on alternative rings
Corollary 2. Let A be a prime non-degenerate alternative algebra on
a field of characteristic 6= 2 containing a idempotent (A need not have
an identity element), and let A′ be an arbitrary alternative algebra. Then
every surjective elementary map (M,M∗) of A× A
′ is additive.
Proof. The result follows directly from Theorem 2 and [1, Theorem 1].
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alternative and Jordan algebras to be prime (Russian). Tr. Mosk. Mat. O.-va 50
(1987) 130-137.
[2] J. C. M. Ferreira and H. Guzzo Jr: Jordan Elementary Maps on Alternative Rings.
Comm. Algebra (2) 42 (2014) 779-794.
[3] I. R. Hentzel, E. Kleinfeld and H. F. Smith: Alternative Rings with Idempotent.
J. Algebra 64 (1980) 325-335.
[4] P. Li and F. Lu: Additivity of Elementary Maps on Rings. Comm. Algebra 32
(2004) 3725-3737.
[5] W. S. Martindale III: When are multiplicative mappings additive? Proc. Amer.
Math. Soc. 21 (1969) 695-698.
[6] M. Slater: Prime alternative rings I. J. Algebra 15 (1970) 229-243.
[7] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov: Rings that
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Novosibirsk, U.S.S.R., Academic Press, New York (1982).
Contact information
Bruno Leonardo
Macedo Ferreira
Universidade Tecnológica Federal do Paraná,
Avenida Professora Laura Pacheco Bastos, 800,
85053-510, Guarapuava, Brazil
E-Mail(s): brunoferreira@utfpr.edu.br
Received by the editors: 20.01.2018
and in final form 25.01.2018.
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