Local multiplications on algebras
Let \(R\) be a commutative ring with identity and let \(A\) be an algebra with identity over \(R\). It is said that an \(R\)-endomorphism \(\varphi\) on \(A\) is LIP (left-ideal-preserving) if \(\varphi(M)\subset M\) for every left ideal \(M\) of \(A\). The algebra \(A\) is said to be SLIP over \(R\) if every LIP \(R\)-endomorphism is a left multiplication by an element of \(A\). The main results proved by the authors are: 1) Suppose \(\{F_i:i\in I\}\) is a collection of two-sided ideals in \(A\) such that \(A/F_i\) is SLIP over \(R\) for every \(i\in I\) and \(\bigcap F_i=\{0\}\), then \(A\) is SLIP over \(R\); 2) Suppose \(R\) is an integral domain. The following are equivalent: (i) \(R\) is infinite. (ii) \(R[S]\) is SLIP over \(R\) for every nonempty set \(S\). (iii) There is a nonempty set \(S\) such that \(R[S]\) is SLIP over \(R\). 3) Suppose \(R\) is an integral domain containing the field of rational numbers \(\mathbb{Q}\) and that, for every positive integer \(n\), \(R\) contains \(n\) distinct \(n\)th roots of unity. Suppose \(G\) is a group that is a subdirect product of torsion groups. Then the group ring \(R[G]\) is SLIP over \(R\).
scientific article; zbMATH DE number 987941