Algebra (ring theory)

In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R.

In this article, all rings are assumed to be unital.

Formal definition

Let R be a commutative ring. An R-algebra is an R-module A together with a binary operation [·, ·]

[\cdot,\cdot]: A \times A\to A

called A-multiplication, which satisfies the following axiom:

Bilinearity:

[\alpha x + \beta y, z] = \alpha [x, z] + \beta [y, z], \quad [z, \alpha x + \beta y] = \alpha [z, x] + \beta [z, y]

for all scalars

\alpha

\beta

in R and all elements x, y, z in A.

Example

Split-biquaternions

Main article: Split-biquaternion

The split-biquaternions are an example of an algebra over a ring that is not a field.

The base ring of the split-biquaternions is the ring of split-complex numbers (or hyperbolic numbers, also perplex numbers), which are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form

x + y j,

where x and y are real numbers. The number j is similar to the imaginary unit i, except that

j² = +1.

A split-biquaternion is a hypercomplex number of the form

q = w + xi + yj + zk \!

where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an element of an eight-dimensional vector space. Considering that it carries a multiplication, this vector space is an algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring. This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of algebras.

Associative algebras

Main article: Associative algebra

If A is a monoid under A-multiplication (it satisfies associativity and it has an identity), then the R-algebra is called an associative algebra. An associative algebra forms a ring over R and provides a generalization of a ring. An equivalent definition of an associative R-algebra is a ring homomorphism $f:R\to A$ such that the image of f is contained in the center of A.

If the ring B is a commutative ring, a simpler, alternative definition is: Given a ring homomorphism $\lambda: A \to B$ we say that B is an A-algebra.^[1]

A ring homomorphism $\rho: A \to B$ shall always map the identity of A to the identity of B. We also say that B/A is an algebra over A given by $\rho$ . Every ring is a $\mathbb{Z}$ -algebra.^[2]

Non-associative algebras

Main article: Non-associative algebra

A non-associative algebra^[3] (or distributive algebra) over a field (or a commutative ring) K is a K-vector space (or more generally a module^[4]) A equipped with a K-bilinear map A × A → A which establishes a binary multiplication operation on A. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.

References

↑ H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
↑ Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
↑ Schafer 1966, Chapter 1.
↑ Schafer 1966, pp.1.