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Let R {\displaystyle R} be a commutative unital ring, M {\displaystyle M} a R {\displaystyle R} -module, and x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} be a sequence of elements in R {\displaystyle R} . We say that the x i {\displaystyle x_{i}} s form a regular sequence on M {\displaystyle M} if the following two conditions hold:
be a commutative unital ring, 
a -module, and 
be a sequence of elements in . We say that the 
s form a regular sequence on if the following two conditions hold:
, is a nonzerodivisor on 