Mappings between the lattices of varieties of submodules

Document Type : Research Paper

Authors

Department of Mathematics, University of Birjand, Birjand, Iran.

Abstract

Let

R

be a commutative ring with identity and

M

be an

R

-module. It is shown that the usual lattice

V (_{R} M)

of varieties of submodules of

M

is a distributive lattice. If

M

is a semisimple

R

-module and the unary operation

^{'}

V (_{R} M)

is defined by

(V (N))^{'} = V (\tilde{N})

, where

M = N \oplus \tilde{N}

, then the lattice

V (_{R} M)

with

^{'}

forms a Boolean algebra. In this paper, we examine the properties of certain mappings between

V (_{R} R)

and

V (_{R} M)

, in particular considering when these mappings are lattice homomorphisms. It is shown that if

M

is a faithful primeful

R

-module, then

V (_{R} R)

and

V (_{R} M)

are isomorphic lattices, and therefore

V (_{R} M)

and the lattice

R (R)

of radical ideals of

R

are anti-isomorphic lattices. Moreover, if

R

is a semisimple ring, then

V (_{R} R)

and

V (_{R} M)

are isomorphic Boolean algebras, and therefore

V (_{R} M)

and

L (R)

are anti-isomorphic Boolean algebras.

Keywords

20.1001.1.23453931.2022.10.1.3.1