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The simplest form of an algebraic expression, a [[Polynomial|polynomial]] containing only one term.
 
The simplest form of an algebraic expression, a [[Polynomial|polynomial]] containing only one term.
  
Like polynomials (see [[Ring of polynomials|Ring of polynomials]]), monomials can be considered not only over a field but also over a ring. A monomial over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647601.png" /> in a set of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647603.png" /> runs through some index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647604.png" />, is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647607.png" /> is a mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647608.png" /> into the set of non-negative integers, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m0647609.png" /> for all but a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476010.png" />. A monomial is usually written in the form
+
Like polynomials (see [[Ring of polynomials|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476011.png" /></td> </tr></table>
+
$$
 +
a x _ {i _ {1}  } ^ {\nu ( i _ {1} ) } \dots
 +
x _ {i _ {n}  } ^ {\nu ( i _ {n} ) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476012.png" /> are all the indices for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476013.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476014.png" /> is called the degree of the monomial in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476015.png" />, and the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476016.png" /> is called the total degree of the monomial. The elements of the ring can be regarded as monomials of degree 0. A monomial with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476017.png" /> is called primitive. Any monomial with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476018.png" /> is identified with the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476019.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476020.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476022.png" />, forms a commutative semi-group with identity. Here the product of two monomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476024.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476025.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476026.png" /> be a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476027.png" />-algebra. Then the monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476028.png" /> defines a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476030.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476031.png" />.
+
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
 
Monomials in non-commuting variables are sometimes considered. Such monomials are defined as expressions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476032.png" /></td> </tr></table>
+
$$
 +
a x _ {i _ {1}  } ^ {\nu ( i _ {1} ) } \dots
 +
x _ {i _ {n}  } ^ {\nu ( i _ {n} ) } ,
 +
$$
  
where the sequence of (not necessarily distinct) indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064760/m06476033.png" /> is fixed.
+
where the sequence of (not necessarily distinct) indices $  i _ {1} \dots i _ {n} $
 +
is fixed.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


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)
How to Cite This Entry:
Monomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monomial&oldid=19244
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article