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Monomial

From Encyclopedia of Mathematics
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The simplest form of an algebraic expression, a polynomial containing only one term.

Like polynomials (see Ring of polynomials), monomials can be considered not only over a field but also over a ring. A monomial over a commutative ring in a set of variables , where runs through some index set , is a pair , where and is a mapping of the set into the set of non-negative integers, where for all but a finite number of . A monomial is usually written in the form

where are all the indices for which . The number is called the degree of the monomial in the variable , and the sum is called the total degree of the monomial. The elements of the ring can be regarded as monomials of degree 0. A monomial with is called primitive. Any monomial with is identified with the element .

The set of monomials over in the variables , , forms a commutative semi-group with identity. Here the product of two monomials and is defined as .

Let be a commutative -algebra. Then the monomial defines a mapping of into by the formula .

Monomials in non-commuting variables are sometimes considered. Such monomials are defined as expressions of the form

where the sequence of (not necessarily distinct) indices is fixed.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
How to Cite This Entry:
Monomial. L.V. Kuz'min (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monomial&oldid=19244
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098