Derivations over amalgamated algebras along an ideal

Abstract

Let $f:A\to B$ be a ring homomorphism and I be an ideal of B. The amalgamated algebra of A and B along I with respect to f is the subring of A × B given by

$$A{\bowtie }^{f}I:=\{(a,f(a)+i)|a\in A,i\in I\}.$$

In this paper, we give a complete description of derivations over $A{\bowtie }^{f}I$. The obtained results cover some recent results over duplication of a ring along an ideal. Furthermore, it’s shown that if A is a prime ring with 1 and d is a nonzero derivation of $A{\bowtie }^{f}I$ such that, for each $x\in A{\bowtie }^{f}I,$ d(x) = 0 or d(x) is invertible in $A{\bowtie }^{f}I,$ then A is a division ring and $A{\bowtie }^{f}I$ is identified to $D[x]/(x^2),$ where D is a division ring, char(D) = 2, $d(D)=\left\{0\right\}$ and $d(x)=1+ax$ for some $a\in Z(D).$

Keywords:

• Amalgamation
• derivation
• homomorphism
• prime ring

2010 Mathematics Subject Classification:

• 16N60
• 46J10
• 16W25
• 16N20