ideal generated by a subset of a ring (original) (raw)

Let X be a subset of a ring R. Let S={Ik} be the collection of all left ideals of R that contain X (note that the set is nonempty since X⊂R and R is an ideal in itself). The intersection

is called the left ideal generated by X, and is denoted by (X). We say that X generates I as an ideal.

Alternatively, we can constructively form the set of elements that constitutes this ideal: The left ideal (X) consists of finite R-linear combinations of elements of X: