(algebra, ring theory) An algebra over a reduced ring.

(algebra) A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.

(algebra, field theory, of a field F) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

(algebra) An algebra over a field; a ring with identity together with an injective ring homomorphism from a field, k, to the ring such that the image of the field is a subset of the center of the ring and such that the image of the field’s unity is the ring’s unity.

(mathematics) A generalization of central simple algebras to R-algebras where R need not be a field.

(algebra) A direct sum with an exterior product of multivector spaces which are all based on a same underlying finite-dimensional vector space. The associative algebra of sums and exterior products of scalars, vectors, blades, multivectors, and hybrid (i.e., non-homogeneous) sums of multivectors of different grades.

(algebra) A bounded distributive lattice with a dual endomorphism (where “dual” means that it satisfies De Morgan’s laws).

(algebra) A linear algebra such that its non-zero vectors form a multiplicative group.

(algebra) Generically, the arithmetic of matrices, involving matrix addition, scalar multiplication of matrices, matrix multiplication, and matrix inversion.

(algebra) polynomial ring