Minimal Faithful Modules over Artinian Rings

Abstract

Let $R$ be a left Artinian ring, and $M$ a faithful left $R$-module such that no proper submodule or homomorphic image of $M$ is faithful.

If $R$ is local, and $\operatorname{socle}(R)$ is central in $R$, we show that $\operatorname{length}(M/J(R)M) + \operatorname{length}(\operatorname{socle}(M))\leq \operatorname{length}(\operatorname{socle}(R))+1$.

If $R$ is a finite-dimensional algebra over an algebraically closed field, but not necessarily local or having central socle, we get an inequality similar to the above, with the length of $\operatorname{socle}(R)$ interpreted as its length as a bimodule, and the summand $+1$ replaced by the Euler characteristic of a graph determined by the bimodule structure of $\operatorname{socle}(R)$. The statement proved is slightly more general than this summary; we examine the question of whether much stronger generalizations are possible.

If a faithful module $M$ over an Artinian ring is only assumed to have one of the above minimality properties -- no faithful proper submodules, or no faithful proper homomorphic images -- we find that the length of $M/J(R)M$ in the former case, and of $\operatorname{socle}(M)$ in the latter, is $\leq\operatorname{length}(\operatorname{socle}(R))$. The proofs involve general lemmas on decompositions of modules.