## Journal of Commutative Algebra

### Anti-homomorphisms between module lattices

Patrick F. Smith

#### Abstract

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

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

Dates
First available in Project Euclid: 19 January 2016

https://projecteuclid.org/euclid.jca/1453211674

Digital Object Identifier
doi:10.1216/JCA-2015-7-4-567

Mathematical Reviews number (MathSciNet)
MR3451356

Zentralblatt MATH identifier
1341.13006

#### Citation

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. https://projecteuclid.org/euclid.jca/1453211674

#### References

• Y. Al-Shaniafi and P.F. Smith, Comultiplication modules over commutative rings, J. Comm. Alg. 3 (2011), 1–29.
• F.W. Anderson and K.R. Fuller, Rings and categories of modules, Springer-Verlag, New York, 1974.
• H. Ansari-Toroghy and H. Farshadifar, The dual notion of multiplication modules, Taiwan. J. Math. 11 (2007), 1189–1201.
• ––––, Comultiplation modules and related results, Honam Math. J. 30 (2008), 91–99.
• ––––, On comultiplication modules, Kor. Ann. Math. 25 (2008), 57–66.
• J. Dauns, Prime modules, J. reine angew Math. 298 (1978), 156–181.
• R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
• C.R. Hajarnavis and N.C. Norton, On dual rings and their modules, J. Alg. 93 (1985), 253–266.
• A. Harmanci, P.F. Smith and Y. Tiras, A characterization of prime submodules, J. Alg. 212 (1999), 743–752.
• I. Kaplansky, Dual rings, Ann. Math. 49 (1948), 689–701.
• C.-P. Lu, Prime submodules of modules, Comm. Math. Univ. Sanct. Paul. 33 (1984), 61–69.
• R.L. McCasland and M.E. Moore, Prime submodules, Comm. Alg. 20 (1992), 1803–1817.
• R.L. McCasland and P.F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math. 23 (1993), 1041–1062.
• W.K. Nicholson and M.F. Yousif, On dual rings, New Zealand J. Math. 28 (1999), 65–70.
• Y. Shaniafi and P.F. Smith, Comultiplication modules over commutative rings II, J. Comm. Alg. 4 (2012), 153–174.
• D.W. Sharpe and P. Vamos Injective modules, Cambr. Tracts Math. Math. Phys. 62, Cambridge University Press, Cambridge, 1972.
• P.F. Smith, Mappings between module lattices, Int. Electr. J. Alg. 15 (2014), 173–195.
• ––––, Complete homomorphisms between module lattices, Int. Electr. J. Alg. 16 (2014), 16–31.