Let R be a ring. A well known result of Wedderburn–Artin states that R is generated by simple (right) R-modules if and only if R is uniquely isomorphic to a finite direct product of matrices over division rings. We show that R is generated by virtually simple (right) R-modules if and only if R is uniquely isomorphic to a finite direct product of matrices over principal right ideal domains. An R-module M is called virtually simple (resp. isoartinian) if every nonzero submodule of M is isomorphic to M (resp. every descending chain of submodules of M terminates up to isomorphism). We characterize projective modules generated by virtually simple modules and we show that an isoartinian module <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M_R$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>R</mi> </msub> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S = End_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mi>E</mi> <mi>n</mi> <msub> <mi>d</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has finite u.dim<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(_SM_R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo stretchy="false">(</mo> <mi>S</mi> </msub> <msub> <mi>M</mi> <mi>R</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (resp. u.dim<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(M_R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>R</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) whenever <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M_R$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>R</mi> </msub> </math></EquationSource> </InlineEquation> is nonsingular semiprime (resp. finitely generated extending).

Rings generated by virtually simple modules

期刊论文
发表日期：

2026年7月27日

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Mohammad Reza Vedadi

摘要

Let R be a ring. A well known result of Wedderburn–Artin states that R is generated by simple (right) R-modules if and only if R is uniquely isomorphic to a finite direct product of matrices over division rings. We show that R is generated by virtually simple (right) R-modules if and only if R is uniquely isomorphic to a finite direct product of matrices over principal right ideal domains. An R-module M is called virtually simple (resp. isoartinian) if every nonzero submodule of M is isomorphic to M (resp. every descending chain of submodules of M terminates up to isomorphism). We characterize projective modules generated by virtually simple modules and we show that an isoartinian module $M_R$ $M_{R}$ with $S = End_R(M)$ $S = E n d_{R} (M)$ has finite u.dim $(_SM_R)$ $(_{S} M_{R})$ (resp. u.dim $(M_R)$ $(M_{R})$ ) whenever $M_R$ $M_{R}$ is nonsingular semiprime (resp. finitely generated extending).