In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, the weak dimension of a
nonzero right
module ''M'' over a
ring ''R'' is the largest number ''n'' such that the
Tor group is
nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n'' exists), and the weak dimension of a left ''R''-module is defined similarly. The weak dimension was introduced by . The weak dimension is sometimes called the flat dimension as it is the shortest length of the
resolution of the module by
flat module
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module (mathematics), module ''M'' over a ring (mathematics), ring ''R'' is ''flat'' if taking the tensor prod ...
s. The weak dimension of a module is, at most, equal to its
projective dimension
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, keeping some of the main properties of free modules. Various equivalent characterizatio ...
.
The weak global dimension of a ring is the largest number ''n'' such that
is nonzero for some right ''R''-module ''M'' and left ''R''-module ''N''. If there is no such largest number ''n'', the weak global dimension is defined to be infinite. It is at most equal to the left or right
global dimension of the ring ''R''.
Examples
*The module
of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s over the ring
of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s has weak dimension 0, but projective dimension 1.
*The module
over the ring
has weak dimension 1, but
injective dimension 0.
*The module
over the ring
has weak dimension 0, but injective dimension 1.
*A
Prüfer domain In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely g ...
has weak global dimension at most 1.
*A
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the eleme ...
has weak global dimension 0.
*A
product of infinitely many
fields has weak global dimension 0 but its global dimension is nonzero.
*If a ring is right
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same.
*The
triangular matrix ring has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian, but not left Noetherian.
References
*
*
Commutative algebra
Ring theory
Homological algebra
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