In
mathematics, a
module over a (not necessarily
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
)
ring is countably generated if it is
generated as a module by a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
subset. The importance of the notion comes from
Kaplansky's theorem (Kaplansky 1958), which states that a
projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
is a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of countably generated modules.
More generally, a module over a possibly
non-commutative ring is projective
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
(i) it is
flat, (ii) it is a direct sum of countably generated modules and (iii) it is a
Mittag-Leffler module. (Bazzoni–Stovicek)
References
*
*
Module theory
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