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 ...

, 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 * Modul ...

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 principa ...

(PID) occur in one form of the

structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finite ...

. If

R

is a PID and

M

a finitely generated

R

-module, then ''M'' is isomorphic to a finite sum of the form ::

M\cong R^r\oplus  \bigoplus_^l R/(q_i) \qquad\textr,l\geq0

:where the

(q_i)

are nonzero primary ideals. 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

q_i

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 se ...

, 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

q_i=p_i^

of irreducible elements. The nonnegative integer

r

is called the ''free rank'' or ''Betti number'' of the module

M

. 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\op ...

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 the ...

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.

References

* Chap.11, p.182. * Chap. III.7, p.153 of Module theory {{Abstract-algebra-stub

See also

References