In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is a foundational subject both for algebraic geometry and for the algebraic theory of numbers.
It is considered that the real founder of the subject, at the time when it was called ideal theory, is David Hilbert. It seems that he thinks about it (around 1900) as an alternative approach to the then fashionable theory of complex functions. This approach follows a certain "line" of thought that considers that the computational aspects are secondary with respect to the structural ones. The additional concept of the module, presented in some way in Kronecker's work, is technically a step forward if we compare it with working always directly in the special case of ideals. This change is attributed to the influence of Emmy Noether.
Given the concept of schema, commutative algebra is thought, understood, reasonably, either as the local theory or as the related theory of algebraic geometry.
The general study of rings without requiring commutativity is known as non-commutative algebra; it is a matter of the theory of rings, of the theory of representation and also of other areas such as Banach's theory of algebras.
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