A note on generalized derivations of prime rings
A note on generalized derivations of prime rings
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关于素环广义导数的注记
DOI：
10.3336/gm.40.1.05
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发表时间：
2005
影响因子：
0.4
通讯作者：
Ivica Gusić
中科院分区：
文献类型：
--
作者：
Ivica Gusić
我们证明素环上的广义导数，作为环中非零理想的同态或反同态，是零映射或恒等映射。令 R 为结合环，令 d 为 R 上的导数（即，R 上的加性函数，对于所有 x, y ∈ R 满足 d(xy) = d(x)y + xd(y)），并令 F : R → R 为与 d 相关的广义导数（即，对于所有 x, y ∈ 满足 F (xy) = F (x)y + xd(y) 的加性函数R）。如果关系 aRb = 0 意味着 a = 0 或 b = 0，对于所有 a，b ∈ R，则 R 是素数。请注意，如果 R 是素环并且 I 是 R 的非零理想，则关系 aIb = 0 意味着 a = 0 或 b = 0，对于所有 a，b ∈ R 在 [R，定理 1.2] 中陈述了以下陈述。假设 R 是 2 自由扭且素数。 (i) 如果 d 6= 0 并且 F 作为 R 中非零理想 I 的同态，则 R 是可交换的。 (ii) 如果 d 6= 0 并且 F 充当 R 中非零理想 I 的反同态，则 R 是可交换的。看来这个声明中的假设是矛盾的。此外，尽管论证巧妙，但结论并不完整。使用类似的论证，我们证明以下内容： 定理 1. 令 R 为结合素环，令 d 为 R 上的任何函数（不一定是导数或加法函数），令 F 为 R 上的任何函数（不一定是加法），对于所有 x，y ∈ R 满足 F (xy) = F (x)y+xd(y)，并令 I 为 R 中的非零理想。2000 年数学学科分类。 16W25。
We show that a generalized derivation on a prime ring, that acts as a homomorphism or an anti-homomorphism on a non-zero ideal in the ring, is the zero map or the identity map. Let R be an associative ring, let d be a derivation on R (i.e. an additive function on R satisfying d(xy) = d(x)y + xd(y) for all x, y ∈ R) and let F : R → R be a generalized derivation associated to d (i.e. an additive function satisfying F (xy) = F (x)y + xd(y) for all x, y ∈ R). We say that R is prime if the relation aRb = 0 implies that a = 0 or b = 0, for all a, b ∈ R. Note that if R is a prime ring and I is a non-zero ideal of R, then the relation aIb = 0 implies that a = 0 or b = 0, for all a, b ∈ R In [R, Theorem 1.2] the following statement is stated. Assume that R is 2-torsion free and prime. (i) If d 6= 0 and F acts as a homomorphism on a non-zero ideal I in R then R is commutative. (ii) If d 6= 0 and F acts as an anti-homomorphism on a non-zero ideal I in R then R is commutative. It seems that the assumptions in this statement are contradictory. Also, despite an ingenious argument the conclusion is incomplete. Using a similar argument we prove the following: Theorem 1. Let R be an associative prime ring, let d be any function on R (not necessary a derivation nor an additive function), let F be any function on R (not necessarily additive) satisfying F (xy) = F (x)y+xd(y) for all x, y ∈ R, and let I be a non-zero ideal in R. 2000 Mathematics Subject Classification. 16W25.