nLab divisible module

Definition

If $R$ is a commutative integral domain. An $R$-module $M$ is divisible if $r M = M$ for all $0\neq r \in R$.

According to Levy 1963, in the case of a general ring $R$, we require this only for $r$ which are regular elements (non-zero-divisors).

For a general torsion theory $(T,F)$ (torsion class, torsionfree class) on $R-mod$ for any ring $R$, we say that $M$ in $R-Mod$ is divisible if $E(M)/M$ is torsionfree where $E(M)$ is the injective hull of $M$. This is equivalent to be $J$-injective module where $J$ is the class of all monomorphisms whose cokernel is torsion. In other words, $M$ is divisible if whenever $B/A$ is torsion then any $A\to M$ can be extended to $B\to M$ along the inclusion $A\hookrightarrow B$.

Similarly, $M$ is $(T,F)$-codivisible if for every epimorphism $B\to A$ whose kernel is torsionfree, every map $M\to A$ can be extended to $B$ (in other words it is $K$-projective where $K$ is the class of all epimorphisms with torsionfree kernel).

Properties

If $R$ is a commutative domain, consider the sum $\sigma M$ of all divisible $R$-submodules of $M$ (that is, the maximal divisible $R$-submodule of $M$). The correspondence $M\to\sigma M$ is functorial, in fact $\sigma$ extends to an additive subfunctor of the identity which is idempotent, satisfies $\sigma(M/\sigma(M)) = 0$, but $\sigma$ is not left exact in general.

Literature

For general torsion theories

• Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Springer Lecture Notes in Math. 177, 1971

• Paul E. Bland, Divisible and codivisible modules, Mathematica Scandinavica 34:2 (1974) 153–161 jstor

• Mark Lawrence Teply, A class of divisible modules, Pacific J. Math. 45:2 (1973) 653–668 doi

A notion of divisibility of left $R$-modules with respect to some class $\Sigma$ of left ideals in a unital ring $R$ is introduced in

• D. Sanderson, A generalization of divisibility and injectivity in modules, Canad. Math. Bull. 8 (1965) 505–513

In the case of commutative domains (studying “conditions, some necessary, some sufficient, for the torsion submodule of a divisible module to be a direct summand”) see

• Eben Matlis, Divisible modules, Proc. Amer. Math. Soc. 11:3 (1960) 385–391 jstor

A version for noncommutative domains

• Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963) 132–151 doi

category: algebra