Examples
- Let R be an integral domain with K its field of fractions. Then every R-submodule of K is a fractional ideal. If R is Noetherian, every fractional ideal arises in this way.
- Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups. These are completely classified by the structure theorem, taking Z as the principal ideal domain.
- Finitely generated modules over division rings are precisely finite dimensional vector spaces.
Read more about this topic: Finitely-generated Module
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)