Regular sequence in a ring

Definition

Suppose $R$ is a commutative unital ring, and $x_{1},x_{2},\ldots ,x_{n}$ is a sequence of elements in $R$. We say that the $x_{i}$s form a regular sequence in $R$ if the following are true:

• $(x_{1},x_{2},\ldots ,x_{n})\neq R$
• $x_{i}$ is not a zero divisor in $R/(x_{1},x_{2},\ldots ,x_{i-1})$

The notion generalizes to that of regular sequence on a module.