Commutative algebra

The Commutative algebra is the branch of mathematics in the field of algebra, which deals with commutative rings and their ideals, modules and algebras. It is fundamental to the fields of algebraic geometry and algebraic number theory. An important example of commutative rings are polynomial rings.

As the founder of commutative algebra can be called David Hilbert. He seems the ideal theory ( so the Commutative algebra was originally called ) to have considered alternative access to numerous issues, which could replace the then dominant theory of functions. In this context it structural aspects were more important than algorithmic; with the growing power of computer algebra systems but have concrete calculations become increasingly important within the commutative algebra. The concept of modules, which goes back to basics to Leopold Kronecker, generalizes the theory of ideals that it contains as a special case. These methods were introduced by Emmy Noether in the commutative algebra and are now indispensable.

The general theory of rings, which are not necessarily commutative, is referred to as a non-commutative algebra.

Usual assumptions

In commutative algebra, the terms module, ring and algebra are commonly used in a narrower sense:

• All modules are unitary: If the unit element of the ring is, then for all elements of the module:

• All rings are unitary and commutative.
• Homomorphisms between rings form elements from one to one elements.
• A sub- ring has the same identity element as the top ring.
• All algebras are unitary, commutative and associative.