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.

References

[1]	S. Lang, "Algebra" , Addison-Wesley (1974)