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2015 Minimal Faithful Modules over Artinian Rings
George M. Bergman
Publ. Mat. 59(2): 271-300 (2015).


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.


Download Citation

George M. Bergman. "Minimal Faithful Modules over Artinian Rings." Publ. Mat. 59 (2) 271 - 300, 2015.


Published: 2015
First available in Project Euclid: 30 July 2015

zbMATH: 1327.13060
MathSciNet: MR3374608

Primary: 13E10 , 16G10 , 16P20
Secondary: 16D60

Keywords: Faithful modules over Artinian rings , length of a module or bimodule , socle of a ring or module

Rights: Copyright © 2015 Universitat Autònoma de Barcelona, Departament de Matemàtiques


Vol.59 • No. 2 • 2015
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