Y. Diers in [Diers, 1992, p.45] defined "a range of objects corresponding to the usual types of commutative ring" for a Zariski category (cf. [Archive]). In this note we show how to define these classical objects for any right unitary category.

Let C be a category.

Definition 1. (a) A difference of an object X is a notation a - b where (a, b) is any pair of parallel morphisms from an object to X; a - b is a zero difference if a = b.
(b) A morphism t: X --> T is called a solution of X (or any difference a - b of X); if T is a terminal object then we say that t is a trivial solution of X.

Definition 2. (a) A solution t of a difference a - b is a zero solution if ta = tb.
(b) A difference a - b of an object is unit if its zero solutions are trivial.
(c) A solution t of a difference a - b is a unit solution if ta - tb is a unit.
(d) A difference a - b is nilpotent if its unit solutions are trivial.
(e) A difference a - b is regular if it has a monomorphic solution.

Remark 3. Suppose a - b is a difference and t is a solution of a - b of an object X.
(a) If a - b is both a unit and a zero then X is terminal.
(b) a - b is a zero (resp. unit, resp. nilpotent) implies that ta - tb is so.

In the following we assume any object of C has a unit.

Definition 4. (a) An object is reduced if any non-zero difference has a non-trivial unit solution (i.e. it has no non-zero nilpotent difference).
(b) A non-terminal object is integral if any two non-zero differences has a common monomorphic unit solution (i.e. any non-zero difference is regular).
(c) A non-terminal object is primary if any two non-nilpotent differences has a common monomorphic unit solution.
(d) A non-terminal object is quasi-primary if any two non-nilpotent differences has a common non-trivial unit solution.
(e) A non-terminal object is simple (i.e. a field) if any non-zero difference is a unit.
(f) A non-terminal object is pseudo-simple if any non-nilpotent difference is a unit.
(g) A non-terminal object is local if the class of non-unit differences has a common non-initial null solution.
(h) A non-terminal object is generic if the class of non-nilpotent differences has a common non-initial unit solution.

Remark 5. (a) The classes of reduced, integral, primary, quasi-primary objects are closed under subobjects.
(b) An object is integral iff it is reduced and primary.
(c) Any simple object is integral; any subobject of a simple object is integral.
(d) Any pseudo-simple is primary and any primary object is quasi-primary; any subobject of a pseudo-simple object is primary..
(e) An object is simple iff it is reduced and a pseudo-simple. 

Remark 6. Suppose U = hom(W, ~) is a representable faithful functor from C to the category of sets. By a U-difference (or W-difference) of an object X we mean a notation a - b, where (a, b) is any pair of elements of U(X). Applying the above methods we obtain the notion of U-reduced, U-prime objects, etc. with respective to U-differences. One can show that an object is reduced (resp. prime, etc.) if it is U-reduced (resp. U-prime, etc.) Thus for a concrete category (C, U) (e.g. an algebraic geometry) it suffices to consider U-differences instead of general differences. 

Theorem 7. Suppose (C, U) is an algebraic geometry.
(a) An object is integral iff any non-zero difference has a monomorphic unit solution (or any two non-zero differences has a common non-trivial unit solution).
(b) An object is primary iff any non-nilpotent difference has a monomorphic unit solution (or any two non-nilpotent differences has a common non-trivial unit solution)..
(c) An object is integral iff it is reduced and quasi-primary.
(d) An object is integral iff it is a subobject of a simple object.
(e) An object is primary iff it is a subobject of a pseudo-simple object.
(f) Any direct product of reduced object is reduced; an object is reduced iff it is a subobject of a direct product of simple objects (see [Categorical Geometry, Chapter 3 - 5] for the proof) .