A vector space equipped with a bilinear product. An example is the space of n × n real matrices, with product (A,B)↦AB. As here, the product need not be commutative. See also banach algebra, division algebra, group algebra.

Elementary algebra relates to the general properties of arithmetic and solutions of polynomial equations involving unknown variables. More advanced algebra focuses on algebraic structures that are usually generalizations or abstractions of more concrete problems. The use of algebra, to some extent, permeates all of mathematics, and algebraic techniques have become important in other areas such as algebraic topology. The word algebra derives from a work al-jabr of Khwārizmī. Viète made significant advances in the use of algebraic notation in the 16th century. See also abstract algebra and linear algebra.