Generalized multiplicative derivations in inverse semirings

Abstract

In this note we consider inverse semirings, i.e. semirings $S$ in which for each $a\in S$ there exists a uniquely determined element $a'\in S$ such that $a+a'+a=a$ and $a'+a+a'=a$. If additionally the commutator $[x,y]=xy+y'x$ satisfies both Jordan identities, then such semirings are called Jacobi semirings. The problem of commutativity of such semirings can be solved by specifying easily verifiable conditions which must be satisfied by the commutator or some additive homomorphisms called derivations, or by a pair of nonzero mappings from $S$ to $S$. We consider the pair $(F,f)$ of nonzero mappings $S\to S$ such that $F(xy)=F(x)y+xf(y)$ for all $x,y\in S$ and determine several simple conditions under which the pair $(F,f)$ of such mappings (called a generalized multiplicative derivation) forces the commutativity of a semiring $S$. We show that semiring will be commutative if the conditions we find are satisfied by the elements of a solid ideal, i.e. a nonempty ideal $I$ with the property that for every $x\in I$ elements $x+x'$ are in the center of $I$. For example, a prime Jacobi semiring $S$ with a solid ideal $I$ and a generalized multiplicative derivation $(F,f)$ such that $a(F(xy)+yx)=0$ for all $x,y\in I$ and some nonzero $a\in S$, is commutative. Moreover, in this case $F(s)=s'$ for all $s\in S$ (Theorem 3.2). A prime Jacobi semiring $S$ with a generalized multiplicative derivation $(F,f)$ is commutative also in the case when $S$ contains a nonzero ideal $I$ (not necessarily solid) such that $a(F(x)F(y)+yx)=0$ for all $x,y\in I$ and some nonzero $a\in S$ (Theorem 3.3). Also prime Jacobi semirings with a non zero ideal $I$ and a nonzero derivation $d$ such that $[d(x),x]=0$ for $x\in I$ are commutative.