Module properties

This module is the right regular module over its ring. (The software uses this to deduce properties about the module directly from its ring.)

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 $S+N=M$.

Satisfies descending chain condition on submodules

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

$End(M_R)$ is a division ring

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

Every injective endomorphism of $M_R$ is invertible.

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

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

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

$M$ is generated by a single element.

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

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

The socle of the module is an essential submodule

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

$M$ is Artinian and Noetherian

$M$ has a finitely generated essential semisimple submodule

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

The module has a finite generating set

The socle is finitely generated

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.

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.

The functor $M\otimes_R -$ is exact.

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

$M$ 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 $P$.

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

Every surjective endomorphism of $M_R$ is invertible

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

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

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

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.)

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

Satisfies the ascending chain condition on submodules.

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

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

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

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

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

$M$ 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.

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

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

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

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

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

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

The module is a direct sum of simple submodules.

$M$ is a direct sum of uniserial submodules

The module has only the two trivial submodules.

The socle of $M$ is a simple submodule.

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

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

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

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

The Jacobson radical of $M$ is a superfluous submodule

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

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

The annihilator of each element is nonzero.

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

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

Any two nonzero submodules have a nonzero intersection.

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