In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, the weak dimension of a
nonzero right
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''M'' over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''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 a
resolution
Resolution(s) may refer to:
Common meanings
* Resolution (debate), the statement which is debated in policy debate
* Resolution (law), a written motion adopted by a deliberative body
* New Year's resolution, a commitment that an individual ma ...
of the module by
flat module
In algebra, a flat module over a ring ''R'' is an ''R''- module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact s ...
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, by keeping some of the main properties of free modules. Various equivalent characteriza ...
.
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 In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
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 (e.g. ). The set of all ra ...
s over the ring
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 finitel ...
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
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Prod ...
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
*
*{{Citation , last1=Năstăsescu , first1=Constantin , last2=Van Oystaeyen , first2=Freddy , author2-link = Fred Van Oystaeyen , title=Dimensions of ring theory , publisher=D. Reidel Publishing Co. , series=Mathematics and its Applications , isbn=9789027724618 , doi=10.1007/978-94-009-3835-9 , mr=894033 , year=1987 , volume=36
Commutative algebra
Ring theory
Homological algebra