In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the elementary divisors of a
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 ...
over a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
(PID) occur in one form of the
.
If
is a
PID and
a finitely generated
-module, then ''M'' is isomorphic to a finite sum of the form
::
:where the
are nonzero
primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
s.
The list of primary ideals is unique up to order (but a given ideal may be present more than once, so the list represents a
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
of primary ideals); the elements
are unique only up to
associatedness
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural sett ...
, and are called the ''elementary divisors''. Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers
of irreducible elements. The nonnegative integer
is called the ''free rank'' or ''Betti number'' of the module
.
The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of
invariant factors The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.
If R is a PID and M a finitely generated R-module, then
:M\cong R^r ...
of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
for ''R''. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
See also
*
Invariant factors The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.
If R is a PID and M a finitely generated R-module, then
:M\cong R^r ...
References
* Chap.11, p.182.
* Chap. III.7, p.153 of
Module theory
{{Abstract-algebra-stub