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Let $( A , \mathfrak m )$ be an $r$-dimensional Noetherian ring (cf. also the section  "Dimension of an associative algebra"  in [[Dimension|Dimension]]). Then there exists an $\mathfrak{m}$-primary ideal generated by $r$ elements (cf., e.g., [[#References|[a1]]], p. 98, [[#References|[a2]]], p. 27). If $x _ { 1 } , \dots , x _ { r }$ generate such an $\mathfrak{m}$-primary ideal, they are said to be a system of parameters of $A$. The terminology comes from the situation that $( A , \mathfrak m )$ is the local ring of functions at a (singular) point on an [[Algebraic variety|algebraic variety]]. The system of parameters $x _ { 1 } , \dots , x _ { r }$ is a regular system of parameters if $x _ { 1 } , \dots , x _ { r }$ generate $\mathfrak{m}$, and in that case $( A , \mathfrak m )$ is a regular local ring.

More generally, if $M$ is a finitely-generated $A$-module of dimension $s$, then there are $y _ { 1 } , \dots , y _ { s } \in \mathfrak { m }$ such that $M / ( y _ { 1 } , \ldots , y _ { s } ) M$ is of finite length; in that case $y _ { 1 } , \dots , y _ { s }$ is called a system of parameters of $M$.

The ideal $( y _ { 1 } , \dots , y _ { s } )$ is called a parameter ideal.

For a semi-local ring $A$ with maximal ideals $m _ { 1 } , \dots , m _ { r }$, an ideal $\frak a$ is called an ideal of definition if

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for some natural number $k$. If $A$ is of dimension $d$, then any set of $d$ elements that generates an ideal of definition is a system of parameters of $A$, [[#References|[a3]]], Sect. 4.9.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Matsumura,   "Commutative ring theory" , Cambridge Univ. Press  (1989)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Nagata,   "Local rings" , Interscience  (1962)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D.G. Nothcott,   "Lessons on rings, modules, and multiplicities" , Cambridge Univ. Press  (1968)</td></tr></table>