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.) 

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 $S+N=M$. 

Artinian 
Satisfies descending chain condition on submodules 

Bass module 
A module is called a Bass module if every proper submodule is contained in a maximal submodule. 

brick 
$End(M_R)$ is a division ring 

clean 
The endomorphism ring of $M$ is a clean ring. Also known as: endoclean. 

coHopfian 
Every injective endomorphism of $M_R$ is invertible. 

coherent 
$M_R$ is finitely generated and its finitely generated submodules are finitely presented. 

continuous 
$M$ is a CS module such that any submodule isomorphic to a direct summand of $M$ is again a summand of $M$. 

CS 
Every nonzero submodule of $M$ is essential in a summand of $M$. 

cyclic 
$M$ is generated by a single element. 

distributive 
For all submodules $A,B,C$ of $M$, $A\cap(B+C)=(A\cap C)+(B\cap C)$. 

divisible (naive) 
For every nonzero $r\in R$, we have $Mr=M$. (Note we are not requiring $R$ to be a domain.) 

essential socle 
The socle of the module is an essential submodule 

faithful 
The annihilator in $R$ of $M$ is the trivial ideal. 

finite composition length 
$M$ is Artinian and Noetherian 

finite uniform dimension 
$M$ has a finitely generated essential semisimple submodule 

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 $\cap_{i\in F} N_i=\{0\}$. 

finitely generated 
The module has a finite generating set 

finitely generated socle 
The socle is finitely generated 

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. 

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. 

flat 
The functor $M\otimes_R $ is exact. 

free 
$M$ is isomorphic to a direct sum of copies of $R_R$. 

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 $P$. 

hollow 
The sum of any two proper submodules is still a proper submodule. 

Hopfian 
Every surjective endomorphism of $M_R$ is invertible 

indecomposable 
The module cannot be expressed as a direct sum of two nonzero submodules. 

injective 
For all modules $A, B$, every exact sequence $M\to A\to B\to 0$ splits 

Jacobson semisimple 
The Jacobson radical $J(M)$ is the zero submodule. 

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 finitelysolvable system of congruence is solvable.) 

local 
$M$ has a maximal submodule which is a small submodule. Said another way, $M$ has a maximum proper submodule containing all other submodules. 

Noetherian 
Satisfies the ascending chain condition on submodules. 

nonsingular 
No nonzero element of $M$ has an essential right annihilator in $R$. 

nonzero socle 
$\mathrm{soc}(M)\neq\{0\}$ 

principally injective 
Any homomorphism from a principal ideal $xR\to M$ extends to a homomorphism $R\to M$. 

projective 
For all modules $A, B$, every exact sequence $0\to A\to B\to M\to 0$ splits 

proper Jacobson radical 
$M$ has a maximal submodule, so that $J(M)$, the intersection of maximal submodules of $M$, is a proper submodule. 

quasicontinuous 
$M$ 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. 

quasiinjective 
Every homomorphism from a submodule $N$ of $M$ into $M$ extends to an endomorphism of $M$. 

quasiprojective 
$M$ is quasiprojective if every homomorphism $f:M\to N$ factors through every surjective homomorphism $g:M\to N$. 

reflexive 
The canonical map of $M\to Hom(M_R,R_R)$ is a bijection. 

semiArtinian 
Every nonzero quotient of $M$ has a nonzero socle. Also known as: Loewy modules 

semiNoetherian 
Every nonzero submodule of $M$ has a maximal submodule. Also known as: max modules, Hamsher modules 

semireflexive 
There exist an injection $M\to \prod_{i\in I} R$ for some number of copies of $R$. Also known as: torsionless modules 

semisimple 
The module is a direct sum of simple submodules. 

serial 
$M$ is a direct sum of uniserial submodules 

simple 
The module has only the two trivial submodules. 

simple socle 
The socle of $M$ is a simple submodule. 

singular 
The annihilator of every element of $M$ is an essential right ideal of $R$ 

strongly indecomposable 
$End(M_R)$ is a local ring. Also known as: endolocal modules 

strongly semiNoetherian 
Every quotient of $M$ is semiNoetherian. (The notion appears as "semiNoetherian" in literature, so this term is invented to distinguish it.) 

subdirectly irreducible 
The intersection of all nonzero submodules of $M$ is nonzero. (Or, $M$ has a simple essential submodule.) Also known as: colocal modules 

superfluous Jacobson radical 
The Jacobson radical of $M$ is a superfluous submodule 

supplemented 
For every submodule $N$ of $M$, there exists a submodule $S$ minimal with the property that $S+N=M$. 

top semisimple 
$M/J(M)$ is a semisimple module 

torsion (naive) 
The annihilator of each element is nonzero. 

torsion (regular element) 
The annihilator of each element of the module contains a regular element of the ring. 

torsionfree 
The only module element annihilated by a regular ring element is $0$. 

uniform 
Any two nonzero submodules have a nonzero intersection. 

uniserial 
The submodules of $M$ are linearly ordered. Also known as: chain modules. 