Monomial

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 $ A $ in a set of variables $ \{ x _ {i} \} $, where $ i $ runs through some index set $ I $, is a pair $ ( a, \nu ) $, where $ a \in A $ and $ \nu $ is a mapping of the set $ I $ into the set of non-negative integers, where $ \nu ( i) = 0 $ for all but a finite number of $ i $. A monomial is usually written in the form

$$ a x _ {i _ {1} } ^ {\nu ( i _ {1} ) } \dots x _ {i _ {n} } ^ {\nu ( i _ {n} ) } , $$

where $ i _ {1} \dots i _ {n} $ are all the indices for which $ \nu ( i) > 0 $. The number $ \nu ( i) $ is called the degree of the monomial in the variable $ x _ {i} $, and the sum $ \sum _ {i \in I } \nu ( i) $ is called the total degree of the monomial. The elements of the ring can be regarded as monomials of degree 0. A monomial with $ a = 1 $ is called primitive. Any monomial with $ a = 0 $ is identified with the element $ 0 \in A $.

The set of monomials over $ A $ in the variables $ \{ x _ {i} \} $, $ i \in I $, forms a commutative semi-group with identity. Here the product of two monomials $ ( a , \nu ) $ and $ ( b , \kappa ) $ is defined as $ ( ab , \nu + \kappa ) $.

Let $ B $ be a commutative $ A $- algebra. Then the monomial $ a x _ {i _ {1} } ^ {\nu ( i _ {1} ) } \dots x _ {i _ {n} } ^ {\nu ( i _ {n} ) } $ defines a mapping of $ B ^ {n} $ into $ B $ by the formula $ ( b _ {1} \dots b _ {n} ) \rightarrow a b _ {1} ^ {\nu ( i _ {1} ) } \dots b _ {n} ^ {\nu ( i _ {n} ) } $.

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

$$ a x _ {i _ {1} } ^ {\nu ( i _ {1} ) } \dots x _ {i _ {n} } ^ {\nu ( i _ {n} ) } , $$

where the sequence of (not necessarily distinct) indices $ i _ {1} \dots i _ {n} $ is fixed.

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