Commutative algebra
The commutative algebra is the branch of mathematics in the field of algebra that deals with commutative rings and their ideals , modules and algebras concerned. It is fundamental to the areas of algebraic geometry and algebraic number theory . Polynomial rings are an important example of commutative rings .
David Hilbert can be named as the founder of commutative algebra . He seems to have viewed ideal theory (that's how commutative algebra was originally called) as an alternative approach to numerous questions that could replace the function theory that was dominant at the time . In this context, structural aspects were more important to him than algorithmic; With the increasing efficiency of computer algebra systems , however, concrete calculations have become increasingly important within commutative algebra. The concept of modules, which can be traced back to Leopold Kronecker , generalizes the theory of ideals, which it contains as a special case. These methods were introduced to commutative algebra by Emmy Noether and are indispensable today.
The theory of general rings that need not be commutative is called noncommutative algebra .
Common assumptions
In commutative algebra, the terms module, ring and algebra are usually used in a narrower sense:
 All modules are unitary : If the single element of the ring is, then applies to all elements of the module:
 All rings are unitary and commutative .
 Homomorphisms between rings map single elements onto single elements.
 A lower ring has the same element as the upper ring.
 All algebras are unitary, commutative and associative .
literature

MF Atiyah , IG MacDonald : Introduction to Commutative Algebra. Oxford 1969, AddisonWesley Publishing Company, Inc. ISBN 0201003619 .
Brief introduction, standard work.  Hideyuki Matsumura: Commutative Algebra. WA Benjamin, New York 1970.
More extensive than Commutative Ring Theory, but out of print.  Oscar Zariski , Pierre Samuel : Commutative Algebra. 2 vols., SpringerVerlag, New York 1975, ISBN 0387900896 .
 Hideyuki Matsumura: Commutative Ring Theory. Cambridge University Press, Cambridge 1989, ISBN 0521367646 .
 Yves Diers: Categories of Commutative Algebras. Oxford University Press, 1992, ISBN 0198535864 .

David Eisenbud : Commutative Algebra with a View Toward Algebraic Geometry. SpringerVerlag, New York 1996, ISBN 0387942696 .
Comprehensive standard work.