Module (mathematics): Difference between revisions

Revision as of 12:13, 17 March 2025

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The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by

:<math> ( r s ) \cdot x = s \cdot ( r \cdot x ), </math>

one gets a right-module, even if the scalars are written on the left. HoweveerHowever, writing the scalars on the left for left-modules and on the right for right modules makes the manipulation of property 3 much easier.

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>