# Module properties

This is the comprehensive list of module properties in the database.

Name Definition % Complete
$R_R$
This module is the right regular module over its ring. (The software uses this to deduce properties about the module directly from its ring.)
100%
amply supplemented
For every submodule $N$ of $M$, and every submodule $T$ such that $T+N=M$, there exists a submodule $S$ contained in $T$ minimal with the property that $S+N=M$.
47.6%
Artinian
Satisfies descending chain condition on submodules
100%
Bass module
A module is called a Bass module if every proper submodule is contained in a maximal submodule.
100%
Bezout
Every finitely generated submodule is cyclic.
66.7%
brick
$End(M_R)$ is a division ring
95.2%
clean
The endomorphism ring of $M$ is a clean ring. Also known as: endoclean.
57.1%
co-Hopfian
Every injective endomorphism of $M_R$ is invertible.
76.2%
coherent
$M_R$ is finitely generated and its finitely generated submodules are finitely presented.
90.5%
continuous
$M$ is a CS module such that any submodule isomorphic to a direct summand of $M$ is again a summand of $M$.
81.0%
CS
Every nonzero submodule of $M$ is essential in a summand of $M$.
85.7%
cyclic
$M$ is generated by a single element.
95.2%
distributive
For all submodules $A,B,C$ of $M$, $A\cap(B+C)=(A\cap C)+(B\cap C)$.
76.2%
divisible (naive)
For every nonzero $r\in R$, we have $Mr=M$. (Note we are not requiring $R$ to be a domain.)
61.9%
essential socle
The socle of the module is an essential submodule
90.5%
faithful
The annihilator in $R$ of $M$ is the trivial ideal.
76.2%
finite composition length
$M$ is Artinian and Noetherian
100%
finite uniform dimension
$M$ has a finitely generated essential semisimple submodule
95.2%
finitely cogenerated
If $\{N_i\mid i\in I\}$ is a collection of submodules such that $\cap_{i\in I} N_i=\{0\}$, then there exists a finite subset $F$ of $I$ such that $\cap_{i\in F} N_i=\{0\}$.
90.5%
finitely generated
The module has a finite generating set
100%
finitely generated socle
The socle is finitely generated
85.7%
finitely presented
There exists a surjective homomorphism $F_R\to M_R$ where the kernel is finitely generated and $F_R$ is a finitely generated free module.
95.2%
finitely related
There is a surjective homomorphism $F_R\to M_R$ where $F_R$ is a free module, and the kernel of this homomorphism is finitely generated.
76.2%
flat
The functor $M\otimes_R -$ is exact.
100%
free
$M$ is isomorphic to a direct sum of copies of $R_R$.
100%
has a projective cover
$M$ has a projective cover when there exists a projective module $P$ and surjective homomorphism $f:P\to M$ such that the kernel of $f$ is a small submodule of $P$.
90.5%
hollow
The sum of any two proper submodules is still a proper submodule.
90.5%
Hopfian
Every surjective endomorphism of $M_R$ is invertible
85.7%
indecomposable
The module cannot be expressed as a direct sum of two nonzero submodules.
100%
injective
For all modules $A, B$, every exact sequence $M\to A\to B\to 0$ splits
85.7%
Jacobson semisimple
The Jacobson radical $J(M)$ is the zero submodule.
71.4%
linearly compact
Given any family of submodules $N_i$ of $M$ indexed by $I$, and elements $m_i$ indexed by the same set, if $\bigcap_{i\in F}(m_i+N_i)\neq \emptyset$ for every finite subset $F$ of $I$, then $\bigcap_{i\in I}(m_i+N_i)\neq \emptyset$. (In other words, every finitely-solvable system of congruence is solvable.)
95.2%
local
$M$ has a maximal submodule which is a small submodule. Said another way, $M$ has a maximum proper submodule containing all other submodules.
95.2%
Noetherian
Satisfies the ascending chain condition on submodules.
100%
nonsingular
No nonzero element of $M$ has an essential right annihilator in $R$.
61.9%
nonzero socle
$\mathrm{soc}(M)\neq\{0\}$
90.5%
principally injective
Any homomorphism from a principal ideal $xR\to M$ extends to a homomorphism $R\to M$.
81.0%
projective
For all modules $A, B$, every exact sequence $0\to A\to B\to M\to 0$ splits
90.5%
$M$ has a maximal submodule, so that $J(M)$, the intersection of maximal submodules of $M$, is a proper submodule.
100%
quasi-continuous
$M$ is a CS module and if $A$, $B$ are summands of $M$ such that $A\cap B=\{0\}$, then $A\oplus B$ is a summand of $M$ also.
81.0%
quasi-injective
Every homomorphism from a submodule $N$ of $M$ into $M$ extends to an endomorphism of $M$.
81.0%
quasi-projective
$M$ is quasi-projective if every homomorphism $f:M\to N$ factors through every surjective homomorphism $g:M\to N$.
71.4%
reflexive
The canonical map of $M\to Hom(M_R,R_R)$ is a bijection.
71.4%
semi-Artinian
Every nonzero quotient of $M$ has a nonzero socle. Also known as: Loewy modules
90.5%
semi-Noetherian
Every nonzero submodule of $M$ has a maximal submodule. Also known as: max modules, Hamsher modules
85.7%
semi-reflexive
There exist an injection $M\to \prod_{i\in I} R$ for some number of copies of $R$. Also known as: torsionless modules
90.5%
semisimple
The module is a direct sum of simple submodules.
100%
serial
$M$ is a direct sum of uniserial submodules
81.0%
simple
The module has only the two trivial submodules.
100%
simple socle
The socle of $M$ is a simple submodule.
90.5%
singular
The annihilator of every element of $M$ is an essential right ideal of $R$
19.0%
strongly indecomposable
$End(M_R)$ is a local ring. Also known as: endolocal modules
100%
strongly semi-Noetherian
Every quotient of $M$ is semi-Noetherian. (The notion appears as "semi-Noetherian" in literature, so this term is invented to distinguish it.)
90.5%
subdirectly irreducible
The intersection of all nonzero submodules of $M$ is nonzero. (Or, $M$ has a simple essential submodule.) Also known as: co-local modules
95.2%
The Jacobson radical of $M$ is a superfluous submodule
100%
supplemented
For every submodule $N$ of $M$, there exists a submodule $S$ minimal with the property that $S+N=M$.
47.6%
top semisimple
$M/J(M)$ is a semisimple module
57.1%
torsion (naive)
The annihilator of each element is nonzero.
47.6%
torsion (regular element)
The annihilator of each element of the module contains a regular element of the ring.
90.5%
torsion-free
The only module element annihilated by a regular ring element is $0$.
100%
uniform
Any two nonzero submodules have a nonzero intersection.
100%
uniserial
The submodules of $M$ are linearly ordered. Also known as: chain modules.
95.2%