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.) |
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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 $S+N=M$. |
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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. |
|
Bezout |
Every finitely generated submodule is cyclic. |
|
brick |
$End(M_R)$ is a division ring |
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clean |
The endomorphism ring of $M$ is a clean ring. Also known as: endoclean. |
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co-Hopfian |
Every injective endomorphism of $M_R$ is invertible. |
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coherent |
$M_R$ is finitely generated and its finitely generated submodules are finitely presented. |
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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 $M$. |
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cyclic |
$M$ is generated by a single element. |
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distributive |
For all submodules $A,B,C$ of $M$, $A\cap(B+C)=(A\cap C)+(B\cap C)$. |
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divisible (naive) |
For every nonzero $r\in R$, we have $Mr=M$. (Note we are not requiring $R$ to be a domain.) |
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essential socle |
The socle of the module is an essential submodule |
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faithful |
The annihilator in $R$ of $M$ is the trivial ideal. |
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finite composition length |
$M$ is Artinian and Noetherian |
|
finite uniform dimension |
$M$ has a finitely generated essential semisimple submodule |
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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 $\cap_{i\in F} N_i=\{0\}$. |
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finitely generated |
The module has a finite generating set |
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finitely generated socle |
The socle is finitely generated |
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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. |
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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. |
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flat |
The functor $M\otimes_R -$ is exact. |
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free |
$M$ is isomorphic to a direct sum of copies of $R_R$. |
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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 $P$. |
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hollow |
The sum of any two proper submodules is still a proper submodule. |
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Hopfian |
Every surjective endomorphism of $M_R$ is invertible |
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indecomposable |
The module cannot be expressed as a direct sum of two nonzero submodules. |
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injective |
For all modules $A, B$, every exact sequence $M\to A\to B\to 0$ splits |
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Jacobson semisimple |
The Jacobson radical $J(M)$ is the zero submodule. |
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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.) |
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local |
$M$ has a maximal submodule which is a small submodule. Said another way, $M$ has a maximum proper submodule containing all other submodules. |
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Noetherian |
Satisfies the ascending chain condition on submodules. |
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nonsingular |
No nonzero element of $M$ has an essential right annihilator in $R$. |
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nonzero socle |
$\mathrm{soc}(M)\neq\{0\}$ |
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principally injective |
Any homomorphism from a principal ideal $xR\to M$ extends to a homomorphism $R\to M$. |
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projective |
For all modules $A, B$, every exact sequence $0\to A\to B\to M\to 0$ splits |
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proper Jacobson radical |
$M$ has a maximal submodule, so that $J(M)$, the intersection of maximal submodules of $M$, is a proper submodule. |
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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. |
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quasi-injective |
Every homomorphism from a submodule $N$ of $M$ into $M$ extends to an endomorphism of $M$. |
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quasi-projective |
$M$ is quasi-projective if every homomorphism $f:M\to N$ factors through every surjective homomorphism $g:M\to N$. |
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reflexive |
The canonical map of $M\to Hom(M_R,R_R)$ is a bijection. |
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semi-Artinian |
Every nonzero quotient of $M$ has a nonzero socle. Also known as: Loewy modules |
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semi-Noetherian |
Every nonzero submodule of $M$ has a maximal submodule. Also known as: max modules, Hamsher modules |
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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 |
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semisimple |
The module is a direct sum of simple submodules. |
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serial |
$M$ is a direct sum of uniserial submodules |
|
simple |
The module has only the two trivial submodules. |
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simple socle |
The socle of $M$ is a simple submodule. |
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singular |
The annihilator of every element of $M$ is an essential right ideal of $R$ |
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strongly indecomposable |
$End(M_R)$ is a local ring. Also known as: endolocal modules |
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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.) |
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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 |
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superfluous Jacobson radical |
The Jacobson radical of $M$ is a superfluous submodule |
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supplemented |
For every submodule $N$ of $M$, there exists a submodule $S$ minimal with the property that $S+N=M$. |
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top semisimple |
$M/J(M)$ is a semisimple module |
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torsion (naive) |
The annihilator of each element is nonzero. |
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torsion (regular element) |
The annihilator of each element of the module contains a regular element of the ring. |
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torsion-free |
The only module element annihilated by a regular ring element is $0$. |
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uniform |
Any two nonzero submodules have a nonzero intersection. |
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uniserial |
The submodules of $M$ are linearly ordered. Also known as: chain modules. |