denoted R-Alg. The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings. Examples The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics. Algebra Any ring A can be considered as a Z-algebra. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore, rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent. Any ring of characteristic n is a (Z/nZ)-algebra in the same way. Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x). Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated, free R-module. In particular, the square n-by-n matrices with entries from the field K form an associative algebra over K. The complex numbers form a 2-dimensional commutative algebra over the real numbers. The quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions). The polynomials with real coefficients form a commutative algebra over the reals. Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}. The free R-algebra on a set E is an algebra of "polynomials" with coefficients in R and noncommuting indeterminates taken from the set E. The tensor algebra of an R-module is naturally an associative R-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor that maps an R-module to its tensor algebra is left adjoint to the functor that sends an R-algebra to its underlying R-module (forgetting the multiplicative structure). The following ring is used in the theory of λ-rings. Given a commutative ring A, let the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series.