The subject's real founder, in the days when it was called ideal theory, should be considered to be David Hilbert. He seems to have thought of it (around 1900) as an alternate approach that could replace the then-fashionable complex function theory. In line with his thinking, computational aspects were secondary to the structural. The additional module concept, present in some form in Kronecker's work, is technically an improvement on working always directly on the special case of ideals. Its adoption is attributed to Emmy Noether's influence.
The general study of rings that are not required to be commutative is known as noncommutative algebra; it is pursued in ring theory, representation theory and in other areas such as Banach algebra theory.
Pages related to commutative algebra include: