Algebra over a commutative ring

As algebra over a commutative ring or algebra (wherein a commutative ring) is defined as an algebraic structure consisting of a module over a commutative ring and an additional, compatible with the module structure multiplication (algebra) consists. In particular, algebra over a commutative ring is a generalization of algebra over a field . ${\displaystyle R}$ ${\displaystyle R}$

general definition

Be a commutative ring, a - module and ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R}$

${\displaystyle \,\cdot \,\colon A\times A\to A}$

a binary operation on called "multiplication". ${\displaystyle A}$

The pair is called " -algebra" if the multiplication is bilinear , i.e. H. for all algebra elements and each ring element applies: ${\displaystyle (A,\cdot )}$ ${\displaystyle R}$ ${\displaystyle \cdot }$ ${\displaystyle x,y,z\in A}$ ${\displaystyle \lambda \in R}$

• ${\displaystyle (x+y)\cdot z=x\cdot z+y\cdot z,}$
• ${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z,}$
• ${\displaystyle \lambda (x\cdot y)=(\lambda x)\cdot y=x\cdot (\lambda y).}$

Initially, neither associativity nor commutativity nor the existence of a neutral element of the algebra multiplication is assumed here.

Algebra Homomorphism

A - algebra homomorphism from to is an R-module homomorphism, for which it also applies that is for all . ${\displaystyle R}$ ${\displaystyle \varphi }$ ${\displaystyle (A,\cdot )}$ ${\displaystyle (B,\cdot )}$ ${\displaystyle \varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)}$ ${\displaystyle a,b\in A}$

Special definition

Be a commutative ring. An algebra is a tuple . There is a unitary ring and a ring homomorphism into the center of . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle (A,\alpha )}$ ${\displaystyle A}$ ${\displaystyle \alpha \colon R\rightarrow Z(A)}$ ${\displaystyle A}$

properties

• Every so defined -algebra can be understood as -algebra according to the general definition by setting the scalar multiplication as . On the other hand, not every algebra according to the general definition can be reduced to one according to the special one. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \lambda a:=\alpha (\lambda )\cdot a}$ ${\displaystyle R}$
• Furthermore, each as defined does algebra as - bimodule be construed virtue . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle r\cdot a\cdot r':=\alpha (r)\cdot a\cdot \alpha (r')}$
• An -algebra is called finite if it is understood as a -module and is finitely generated. ${\displaystyle R}$ ${\displaystyle R}$
• An -algebra is called finitely generated if there is a surjective algebra homomorphism for it . ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle n\geq 0}$ ${\displaystyle R[X_{1},\ldots ,X_{n}]\longrightarrow A}$

Algebra Homomorphism

For this special definition of an R-algebra, an - algebra homomorphism from to is defined as a ring homomorphism from to , for which is also true . ${\displaystyle R}$ ${\displaystyle \varphi }$ ${\displaystyle (A,\alpha )}$ ${\displaystyle (B,\beta )}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \varphi \circ \alpha =\beta }$

Examples

• Each ring is an algebra, an algebra above the commutative ring of integers . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$
• Each commutative ring is an algebra about itself.
• The polynomial ring over a ring is a finitely generated, but not finite algebra (unless it is the zero ring ). ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

literature

• Serge Lang : Algebra  (= Graduate Texts in Mathematics 211 ), (Rev. 3rd ed.). Edition, Springer, New York 2002, ISBN 0-387-95385-X .
• Michael Francis Atiyah , Ian Macdonald : Introduction to Commutative Algebra  (= Addison-Wesley Series in Mathematics). Westview Press, University of Oxford 1969, ISBN 9780201407518 .