in the category of modules The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules. Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map . The associativity then refers to the identity: From ring homomorphisms An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism whose image lies in the center of A, we can make A an R-algebra by defining for all r ∈ R and x ∈ A. If A is an R-algebra, taking x = 1, the same formula in turn defines a ring homomorphism whose image lies in the center. If a ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism . The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms R → A; i.e., commutative R-algebras and whose morphisms are ring homomorphisms A → A that are under R; i.e., R → A → A is R → A (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R. How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: generic matrix ring. Algebra homomorphisms A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, is an associative algebra homomorphism if The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes