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- W2321928704 abstract "It is well known and not too difficult to show (see [3 ]) that for a left (right) vector space over a division ring, there exists a basis, and all bases have the same cardinality. The same fact about bases is no longer true in a general module over an arbitrary ring. Even if it has bases, in case these bases are finite they do not necessarily have the same cardinality. Everett in [1] and Leavitt in [4] have constructed examples. Consider the following simpler example. Let V be a left vector space over a division ring D, with a countable infinite basis { as }. Let R be the ring of endomorphisms of V into V, where addition is the obvious one and product is defined as composition of mappings. R is a left module over itself, and has a basis of a single element namely the identity endomorphism. But {u1, u2} also form a basis, where we define for all n = 1, 2," @default.
- W2321928704 created "2016-06-24" @default.
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- W2321928704 date "1964-09-01" @default.
- W2321928704 modified "2023-09-24" @default.
- W2321928704 title "Note on Modules" @default.
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- W2321928704 doi "https://doi.org/10.1080/0025570x.1964.11975532" @default.
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