Module (mathematics): Difference between revisions

Revision as of 16:54, 28 April 2022

Revision as of 17:06, 28 April 2022

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Authors who do not require rings to be [[unital algebra|unital]] omit condition 4 in the definition above; they would call the structures defined above "unital left ''R''-modules". In this article, consistent with the [[glossary of ring theory]], all rings and modules are assumed to be unital.<ref name="DummitFoote">{{cite book | title=Abstract Algebra | publisher=John Wiley & Sons, Inc. |author1=Dummit, David S. |author2=Foote, Richard M. |name-list-style=amp | year=2004 | ___location=Hoboken, NJ | isbn=978-0-471-43334-7}}</ref>

An (''R'',''S'')-[[bimodule]] is an abelian group together with both a left scalar multiplication · by elements of ''R'' and a right scalar multiplication *∗ by elements of ''S'', making it simultaneously a left ''R''-module and a right ''S''-module, satisfying the additional condition <math> {{nowrap|1=(''r'' \cdot· ''x'') \ast∗ ''s'' = ''r'' \cdot⋅ ( ''x'' \ast∗ ''s '') </math>}} for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.

If ''R'' is [[commutative ring|commutative]], then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.

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*If ''R'' is a ring, we can define the [[opposite ring]] ''R''^op which has the same [[underlying set]] and the same addition operation, but the opposite multiplication: if {{nowrap|1=''ab'' = ''c''}} in ''R'', then {{nowrap|1=''ba'' = ''c''}} in ''R''^op. Any ''left'' ''R''-module ''M'' can then be seen to be a ''right'' module over ''R''^op, and any right module over ''R'' can be considered a left module over ''R''^op.

* [[Glossary of Lie algebras#Representation theory|Modules over a Lie algebra]] are (associative algebra) modules over its [[universal enveloping algebra]].

*If ''R'' and ''S'' are rings with a [[ring homomorphism]] {{nowrap|''φ'' : ''R'' → ''S''}}, then every ''S''-module ''M'' is an ''R''-module by defining {{nowrap|1=''rm'' = ''φ''(''r'')''m''}}. In particular, ''S'' itself is such an ''R''-module.

== Submodules and homomorphisms ==

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; Indecomposable: An [[indecomposable module]] is a non-zero module that cannot be written as a [[direct sum of modules|direct sum]] of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. [[uniform module]]s).

; Faithful: A [[faithful module]] ''M'' is one where the action of each {{nowrap|''r'' ≠ 0}} in ''R'' on ''M'' is nontrivial (i.e. {{nowrap|''r'' ⋅ ''x'' ≠ 0}} for some ''x'' in ''M''). Equivalently, the [[annihilator (ring theory)|annihilator]] of ''M'' is the [[zero ideal]].

; Torsion-free: A [[torsion-free module]] is a module over a ring such that 0 is the only element annihilated by a regular element (non [[zero-divisor]]) of the ring, equivalently <math>{{nowrap|1=''rm'' = 0</math>}} implies <math>{{nowrap|1=''r'' = 0</math>}} or <math>{{nowrap|1=''m'' = 0</math>}}.

; Noetherian: A [[Noetherian module]] is a module which satisfies the [[ascending chain condition]] on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.

; Artinian: An [[Artinian module]] is a module which satisfies the [[descending chain condition]] on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

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* [[Module (model theory)]]

* [[Module spectrum]]

* [[Annihilator (ring theory)|Annihilator]]

== Notes ==