Module (mathematics): Difference between revisions

Revision as of 23:57, 10 June 2005

Revision as of 18:03, 13 June 2005

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For all ''r'',''s'' in ''R'', ''x'',''y'' in ''M'', we have

# ''r''(''x''+''y'') = ''rx''+''ry''

# (''r''+''s'')''x'' = ''rx''+''sx''

# (''rs'')''x'' = ''r''(''sx'')

# (''r''+''s'')''x'' = ''rx''+''sx''

# ''r''(''x''+''y'') = ''rx''+''ry''

# 1''x'' = ''x''

Usually, we simply write "a left ''R''-module ''M''" or _''R''''M''. A <b>right ''R''-module</b> ''M'' or ''M''_''R'' is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' &times; ''R'' &rarr; ''M'', and the above axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''.

Some authors omit condition 4 for the general definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital, and all rings are assumed to have a one.

SomeAuthors authorswho do not require rings to be [[unital]] omit condition 4 forin the generalabove definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all ring and modules are assumed to be unital, and all rings are assumed to have a one.

A <b>right ''R''-module</b> ''M'' or ''M''_''R'' is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' &times; ''R'' &rarr; ''M'', and the above three axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''.

A [[bimodule]] is a module which is both a left module and a right module.

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==See also==

* [[Modulevector (model theory)space]]

* [[algebra (ring theory)]]

* [[module (model theory)]]

==References==