Anti-homomorphisms between module lattices
In this paper the author has proved the properties of the mapping \(\alpha:\mathcal L(R)\to\mathcal L(_RM)\) defined by setting for each ideal \(B\) of \(R\), \(\alpha(B)\) to be the submodule of \(M\) consisting of all elements \(m\) in \(M\) with \(Bm=0\) and the mapping \(\beta:\mathcal L(_RM)\to\mathcal L(R)\) defined by \(\beta(N)\) is the annihilator in \(R\) of \(N\), for each submodule \(N\) of \(M\), between the lattice \(\mathcal L(R)\) of ideals of a commutative ring \(R\) and the lattice \(\mathcal L(_RM)\) of submodules of an \(R\)-module \(M\), in particular when these mappings are lattice anti-homomorphisms. A mapping \(\varphi\) from a lattice \(L\) to a lattice \(L'\) is an anti-homomorphism, if \(\varphi(a\vee b)=\varphi(a)\wedge\varphi(b)\) and \(\varphi(a\wedge b)=\varphi(a)\vee\varphi(b)\) for all \(a,b\in L\). The following results are proved. The following statements are equivalent for a module \(M\) over a general ring \(R\). (i) Every homomorphic image of \(M\) is a \(\beta\)-module. (ii) Every submodule of \(M\) is a \(\mu\)-module. (iii) \(R=(L:_RN)+(N:_RL)\) for all submodules \(L\) and \(N\) of \(M\). Let \(R\) be any ring, and let an \(R\)-module \(M=M_1\oplus\dots\oplus M_k\) be the direct sum of submodules \(M_i (1\leq i \leq k)\) for some positive integer \(k\). Then \(M\) is a \(\beta\)-module if and only if \(M_i\) is a \(\beta\)-module for each \(1\leq i\leq k\) and \(R=\mathrm{ann}R(M_i)+\mathrm{ann}R(M_j)\) for all integers \(1\leq i<j\leq k\). Let \(R\) be a ring, and let \(M\) denote the \(R\)-module \(\bigoplus_{P\in V(0)} (R/P)\). Then the following statements are equivalent: (i) \(R\) is von Neumann regular. (ii) The mapping \(\alpha:\mathcal L(R)\to\mathcal L(_RM)\) is an anti-monomorphism. (iii) The mapping \(\alpha:\mathcal L(R)\to\mathcal L(_RM)\) is an injection.
scientific article; zbMATH DE number 6533981
scientific article