Content deleted Content added
→Formal definition: clarification; see talk |
m →Submodules and homomorphisms: manually size braces and | divider when using set-builder notation with a summation inside |
||
| (One intermediate revision by one other user not shown) | |||
Line 28:
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.
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>
Line 55:
Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]] of ''M''. Then ''N'' is a '''submodule''' (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product {{nowrap|''r'' ⋅ ''n''}} (or {{nowrap|''n'' ⋅ ''r''}} for a right ''R''-module) is in ''N''.
If ''X'' is any [[subset]] of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be <math display="inline">\langle X \rangle = \,\bigcap_{N\supseteq X} N</math> where ''N'' runs over the submodules of ''M'' that contain ''X'', or explicitly <math display="inline">\
The set of submodules of a given module ''M'', together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [[Lattice (order)|lattice]] that satisfies the '''[[modular lattice|modular law]]''':
| |||