Complete homomorphisms between module lattices
Let \(R\) be a commutative ring with identity and let \(M\) be a unital \(R\) -module, throughout. A ring \(R\) with Jacobson radical \(J\) is called a semiperfect ring in case \(R/J\) is semiprime Artinian allowing idempotents to lift modulo \(J,\) page 303 of \textit{F. W. Anderson} and \textit{K. R. Fuller}'s [Rings and categories of modules. Berlin: Springer-Verlag (1974; Zbl 0301.16001)]. An \(R\)-module \(M\) is called faithful if \(\mathrm{ann}_{R}(M)=(0)\) and \(M\) is a multiplication module if for every \(R\)-submodule \(N\) of \(M\) there is an ideal \(I\) of \(R\) such that \(N=IM.\) Denote by \(\mathcal{L(}_{R}M)\) the lattice of \(R\)-submodules of \(M.\) Regarding \(R\) as an \(R\)-module we can consider the lattice \(\mathcal{L(} _{R}R)\) of ideals of \(R\). Usually \(\mathcal{L(}_{R}\mathcal{R})\) is denoted by \(\mathcal{L(R}).\) Recall that for \(A,B\) in \(\mathcal{L(}_{R}M),A\vee B=A+B \) and \(A\wedge B=A\cap B.\) A lattice \(\mathcal{L}\) is said to be complete if for every nonemplty subset \(S\) of \(\mathcal{L}\) the least upper bound \(\vee (S)\) and the greatest lower bound \(\wedge (S))\) exist and are in \(\mathcal{L} \). Indeed \(\mathcal{L(}_{R}M)\) and \(\mathcal{L(R})\) are complete lattices.NEWLINENEWLINEGiven two lattices \(\mathcal{L}\), \(\mathcal{L}^{\prime }\) a map \(\varphi : \mathcal{L}\rightarrow \mathcal{L}^{\prime }\) is called a homomorphism if for all \(a,b\in \mathcal{L}\) we have \(\varphi (a\vee b)=\varphi (a)\vee \varphi (b)\) and \(\varphi (a\wedge b)=\varphi (a)\wedge \varphi (b).\) If \( \mathcal{L}\) and \(\mathcal{L}^{\prime }\) are both complete a homomorphism \( \varphi :\mathcal{L}\rightarrow \mathcal{L}^{\prime }\) is is called a complete homomorphism if for every nonempty subset \(S\) of \(\mathcal{L}\) we have \(\varphi (\vee (S))=\vee (\varphi (S))=\vee \{\varphi (x):x\in S\}\) and \(\varphi (\wedge (S))=\wedge (\varphi (S))=\wedge \{\varphi (x):x\in S\}.\) The paper under review is essentially to do with complete homomorphisms \( \varphi :\mathcal{L(}R)\rightarrow \mathcal{L(}_{R}M).\) As the results are well described in the abstract we copy them here: ``We prove that the mapping \(\lambda \) from \(\mathcal{L}(R)\) to \(\mathcal{L}(_{R}M)\) defined by \(\lambda (B)=BM\) for every ideal \(B\) of \(R\) is a complete homomorphism if \(M\) is a faithful multiplication module. A ring \(R\) is semiperfect (respectively, a finite direct sum of chain rings) if and only if this mapping \(\lambda : \mathcal{L}(R)\rightarrow \mathcal{L}(_{R}M)\) is a complete homomorphism for every simple (respectively, cyclic) \(R\)-module \(M\). A Noetherian ring \(R\) is an Artinian principal ideal ring if and only if, for every \(R\)-module \(M\), the mapping \(\lambda :L(R)\rightarrow L(RM)\) is a complete homomorphism.''NEWLINENEWLINEReviewer's remark: This paper is a continuation of the author's earlier article [Int. Electron. J. Algebra 15, 173--195 (2014; Zbl 1302.13018)] which should be read before the article under review, for a more satisfying reading.
scientific article; zbMATH DE number 6336952