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Undid good faith revision 1102102031 by Concinnitas43 (talk). "Wherein" is an English word and is what was intended. But I've replaced it by "in which", which is more common language. |
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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'' which contain ''X'', or explicitly <math display="inline">\left\{\sum_{i=1}^k r_ix_i \mid r_i \in R, x_i \in X\right\}</math>, which is important in the definition of tensor products.<ref>{{Cite web|url=http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf|title=ALGEBRA II: RINGS AND MODULES|last=Mcgerty|first=Kevin|date=2016}}</ref>
The set of submodules of a given module ''M'', together with the two binary operations + and ∩, forms a [[Lattice (order)|lattice]] which satisfies the '''[[modular lattice|modular law]]''':
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