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Mathematics Prelims
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Studying math.
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Mathematics Prelims
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2026-03-05 20:21:29
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Mathematics Prelims | Studying math. Mathematics Prelims December 1, 2009 Submodules and Homomorphisms Filed under: Abstract Algebra , Algebra — cjohnson @ 8:57 pm Just as a subgroup is a group within a group or a subfield is a field within a field, a submodule is a module within a module. That is, if is an -module, we say that is a submodule of if is a subgroup of and is closed under multiplication by elements of . This can be summarized by saying that for every and every , the following two properties hold. In the case of groups, we could only form a quotient group if the subgroup we were modding out by as sufficiently “nice” (i.e., was a normal subgroup); likewise, in the case of rings we required that a subring be nice (an ideal) in order to form a quotient ring. With modules however, we can always form the quotient module with a submodule. We can do this since, as is an abelian group under addition, all subgroups are normal and we can form the ...
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