Journal of Commutative Algebra

Anti-homomorphisms between module lattices

Patrick F. Smith

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We examine the properties of certain mappings 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 considering when these mappings are lattice anti-homomorphisms. The mappings in question are the mapping $\alpha : \mathcal{L}(R) \rightarrow \mathcal{L}(_RM)$ 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 $\beta : \mathcal{L}(_RM) \rightarrow \mathcal{L}(R)$ defined by $\beta(N)$ is the annihilator in $R$ of $N$, for each submodule $N$ of $M$.

Article information

J. Commut. Algebra, Volume 7, Number 4 (2015), 567-592.

First available in Project Euclid: 19 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06A30 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13A15: Ideals; multiplicative ideal theory 13C12: Torsion modules and ideals 13C99: None of the above, but in this section

Lattice anti-homomorphism commutative ring Dedekind domain chain ring comultiplication module von Neumann regular ring


Smith, Patrick F. Anti-homomorphisms between module lattices. J. Commut. Algebra 7 (2015), no. 4, 567--592. doi:10.1216/JCA-2015-7-4-567.

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