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 Categorical Concepts in Commutative Algebra   (based on [Unitary Categories]) Consider a commutative ring R with zero 0 and identity 1. Let Z[T] be the polynomial ring over the ring Z of integers. The following basic concepts in Commutative Algebra can be defined categorically: Element of a Ring: (algebraic definition) An element a of a ring R; (categorical definition) A homomorphism t: Z[T] --> R with t(T) = a.  Unit: (algebraic definition) An element a of R is a unit iff there is an element a-1 such that a-1a = 1 (or it generates the largest ideal of R). (categorical definition) An element a of R is a unit iff for any ring homomorphism f: R --> S, f(a) = 0 implies that  S is the zero ring 0.  Nilpotent Element: (algebraic definition) An element a of a ring is nilpotent iff an = 0 for n. (categorical definition) An element a of a ring is nilpotent iff for any ring homomorphism f: R --> S, f(a) is a unit implies that S is the zero ring 0. Ideal: (algebraic definition) A subset a of R is an ideal if it is closed under the operations - and  with any element of R. (categorical definition) A subset a of R is an ideal if it is the kernel of a homomorphism (i.e. the inverse image of the zero element under a ring homomorphism). Radical ideal: (algebraic definition) An ideal a is radical iff for any element a of R, an is in a implie that a is in a. (categorical definition) An ideal a is radical iff it contains any element a such that, if f: R --> S is any ring homomorphism with f(a) a unit, then f(a) generates the largest ideal of S. Prime ideal: (algebraic definition) A non-zero ideal a is prime iff for any element a, b of R, ab is in R iff a or b is in a. (categorical definition) A non-zero ideal a is prime iff it is radical and universal irreducible (i.e. an ideal a is irreducible if the intersection of b and c is in a implies that b or c is in a; a. is called universaly irreducible if its inverse image under any homomorphism is irreducible). Remark. (a) Z[T] is a generator for the category CRing of commutative rings (with unit). (b) Any zero homomorphism 0R: Z[T] --> R factors through the zero homomorphism 0Z: Z[T] --> Z; the codiagonal homomorphism from the sum Z[T, T'] of Z[T] to Z[T] is the pushout of 0Z: Z[T] --> Z along substraction operator - : Z[T] --> Z[T, T'] sending T to T - T'.  (c) Z is a terminal object of CRing. The homomorphism 0Z: Z[T] --> Z is a generic coequalizer in the following sense: Definition. A generic coequalizer for a category A is a map e: E --> Z where E is a generator of A and Z is a terminal object of A such that the codiagonal map of the sum E + E to E is the pushout of e along a map E --> E + E.  Remark. Suppose A has a generic coequalizer g: E --> Z. (a) If A has arbitrary cointersection of coequalizers then any coequalizer is a cointersection of pushouts of g: E --> Z (because this is so for the codiagonal map of the sum E + E to E) (b) For any object X consider the set E(X) = hom(E, X). Take the composite of g: E --> Z with the map Z --> X as the zero map 0X: E --> X. For any a: E --> X and f: X --> Y write f(a) = 0 if fa = 0Y. Next define a subset of E(X) to be an ideal on X if it is an intersection of the kernels f-1(0) for some maps f with domain X. Just as in the case of commutative rings one can define the notions of  radical ideals and prime ideals, etc.