ringslinear

Ringslinear is a term sometimes used to describe a ring that carries a compatible linear structure, effectively a K-algebra. In this setup there is a field K, the ring R is given the structure of a K-vector space, and the ring multiplication is bilinear over K: for all a, b in R and α, β in K, (αa)(βb) = αβ(ab). Equivalently, R becomes a K-algebra with a K-bilinear multiplication map and, in many contexts, a distinguished multiplicative identity.

In a ringslinear, the scalar action of K interacts with ring operations in the expected way: addition and scalar multiplication satisfy the distributive and associativity conditions, while multiplication respects the K-vector space structure. This makes R simultaneously a ring and a K-vector space with a compatible algebra structure.

Examples include the polynomial ring k[x] over a field k, the matrix ring M_n(k), and the algebra of continuous functions C(X, k) on a topological space X. Each of these is a ring equipped with a K-vector space structure that makes multiplication bilinear over k.

Morphisms between ringslinear (often called K-algebra homomorphisms) are maps that preserve both the ring operations and the K-linear structure. Concretely, they are functions that are both ring homomorphisms and K-linear maps.

Notes: the term ringslinear is not as standard as K-algebra, which is the conventional designation in most texts. When encountered, ringslinear typically refers to the same idea: a ring equipped with a compatible linear (vector space) structure over a field. See also algebra over a field, K-algebra, bilinear map, and representation.