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 non-commutative 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: ${\displaystyle 1}$

${\displaystyle 1\cdot m=m}$

• 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, Addison-Wesley Publishing Company, Inc. ISBN 0-201-00361-9 .
  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., Springer-Verlag, New York 1975, ISBN 0-387-90089-6 .
• Hideyuki Matsumura: Commutative Ring Theory. Cambridge University Press, Cambridge 1989, ISBN 0-521-36764-6 .
• Yves Diers: Categories of Commutative Algebras. Oxford University Press, 1992, ISBN 0-198-53586-4 .
• David Eisenbud : Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag, New York 1996, ISBN 0-387-94269-6 .
  Comprehensive standard work.