Given two elements a, b in the Euclidean ring R their least common multiple cЄR is an element in R such that a│c and b│c and such that whenever a│x and b│x for xЄR then c│x

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Modern Algebra
Ring Theory (XVII)
Euclidean Ring
Least Common Multiple

By:- Thokchom Sarojkumar Sinha

Given two elements in the Euclidean ring their least common multiple is an element in such that
and and such that whenever and for then . Prove that any two elements in the
Euclidean ring have a least common multiple in .

Solution:- Let be any two elements in the Euclidean ring . Let be the least common multiple such that and
and whenever and for then .

To prove that .

We have,
and ,
also and and
then .
For and for then .
, where ----------------------------------(1)

Let and , where and are prime elements.

From (1),

each of and is equal to the product of some and .
, where each and may or may not be distinct.

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Note:- Least Common Multiple
Let , where is a Euclidean ring. The least common multiple of and is an element
in such that
(1) and
(2) whenever and , then , for .
The least common multiple of and is denoted by .

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