The notion of `generation', known for monoids,
is similar for rings.
Let R be a ring.
Theorem
H
C
H
If C
is a collection of subrings of R, then
H
C
H
is also a subring of R.
This shows that the smallest subring containing a given set exists (it is the indicated intersection). Therefore, the definition below makes sense.
Definition
Let D be a subset of R. By <D> we denote the smallest subring of R that contains D, the subring generated by D.
Explicitly, this ring consists of all finite sums of products of
elements from D or -D = {-d | d in D}
including 0 (the empty sum) and 1 (the empty product). If a ring
can be generated by finitely many elements, it is called
finitely generated.