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 0_R: Z[T] --> R factors through the zero homomorphism 0_Z: Z[T] --> Z; the codiagonal homomorphism from the sum Z[T, T'] of Z[T] to Z[T] is the pushout of 0_Z: 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 0_Z: 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 0_X: E --> X. For any a: E --> X and f: X --> Y write f(a) = 0 if f_a = 0_Y. 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.