When an affine algebraic scheme Spec(R) can be written as the “categorical union” of two closed proper subschemes Spec(R/I) and Spec(R/J), we say that it is reducible; the term “categorical union” indicates the smallest closed subscheme Spec(R/K) of Spec(R) such that Spec(R/I)(A) Spec(R/J)(A) is contained in Spec(R/K)(A) for every ring A in Spec. Assuming the result that the solutions in finite rings determine the ideal, we see that reducibility is equivalent to the condition that (0) = I capJ with neither I nor J the zero ideal. Thus we can define a ring R to be irreducible if (0) cannot be written as the intersection of two non-zero ideals; an ideal I in R is irreducible if R/I is irreducible. From the Noetherian-ness of the ring R it follows that the zero ideal is a finite intersection of irreducible ideals.
There are some problems with this decomposition. First of all there is the lack of unique-ness. Secondly, the condition that an ideal is irreducible seems somewhat difficult to characterise in terms of elements. Finally, (and very importantly) the notion of categorical irreducibility seems too strong a condition to impose. For example, even the scheme Spec(k[x,y]/(x2,xy,y2) is reducible with this definition; this seems rather counter-intuitive.
Another way to decide if a scheme is irreducible is as follows. Let f and g be non-zero functions with fg = 0, then f = 0 and g = 0 would define a decomposition of the space unless f (or g is nilpotent). This corresponds to the condition that (0) is a primary ideal; an ideal Q is primary if whenever fg lies in Q and g does not lie in Q, some power of f lies in Q.
1.1. Irreducibles are primary. Let R be an irreducible ring and a be in R. If a is not nilpotent then Ra = R[T]/(aT - 1) is a non-zero ring. The kernel J of the natural homomorphism R Ra is a proper ideal. Let J = (x1,...,xr) (since R is Noetherian) and let n be such that anxi = 0 for all i. If y is an element of R such that amy = 0 for some m, then y lies in J and so xny = 0. Now, suppose that c lies in Ran Ann(a); we have c = dan and ca = dan+1 = 0. But then d lies in J and so c = dan = 0. Since we have written (0) as an intersection of two ideals and Ran is non-zero we see that Ann(a) = 0. So we have shown that every non-nilpotent element of R is not a zero-divisor. Let P be the ideal consisting of nilpotent elements. If xy lies in P then xnyn is zero for some n, so either xn = 0 or y is a zero divisor, hence nilpotent. Thus P is a prime ideal. Moreover, if xy = 0 and x does not lie in P then x is not nilpotent and so we must have y = 0. In other words, (0) is a P-primary ideal.
1.2. Irredundant primary decomposition. If P is a prime ideal in R, then the intersection of finitely many P-primary ideals in R is again a P-primary ideal. Thus, given an expression iIi = (0) of (0) as an intersection of irreducible ideals we can bunch together all the Ii’s which are primary for the same prime ideal. This way we obtain an expression jQj = (0), where Qj are Pj-primary for some finite collection of distinct prime ideals Pj of R. Now, we can further assume that (0) jkQk for any j else we can shorten the above expression. Such an expression of (0) as an intersection of primary ideals that cannot be shortened further is called an irredundant primary decomposition of (0). Consider the ring R = k[x,y] = k[X,Y ]/(X2,XY ). We see that (0) is the intersection of the ideals RxRyn for any n. This shows that we cannot home for uniqueness of the primary decomposition. However, we will see that some form of uniqueness can be proved.
1.3. Annihilators which are primary. Let Qi = (0) be an irredundant primary decomposition of (0) and let Pi be the prime ideal which is the radical of Qi. By irredundancy, the intersection jkQj is non-zero; let c be an element of this intersection. Now suppose xy lies in Ann(c) so that xyc = 0. Thus xyc lies in Qk, if x does not lie in Pk then yc lies in Qk and as a multiple of c it lies in jkQj as well. But then yc lies in (0) = jQj and so is 0; in other words y lies in Ann(c). We have shown that Ann(c) is Pj-primary. It is also clear that Ann(c) contains Qj. For any d not in Ann(c) consider the proper ideal Ann(dc). If xy lies in this ideal then xyd lies in Ann(c); so if x is not in Pj, then yd lies in Ann(c) (by the Pj-primality of this ideal) and so y lies in Ann(dc). In other words Ann(dc) is also Pj-primary. On the other hand, if b lies in Pj there is a positive integer n such that bn+1 lies in Ann(c) but bn does not. Then b lies in Ann(bnc). Thus the maximal ideal of the form Ann(dc) must contain and thus equal Pj. We have therefore shown that prime ideals that occur in an irredundant primary decomposition of (0) in the ring R are all annihilators of some element of R.
1.4. Associated primes. The associated primes of a ring R are the prime ideals P which are of the form Ann(c) for some c in R. Suppose P is such a prime. Now consider an irredundant primary decomposition jQj = (0). Among the Qj let A be the collection of those j for which c does not lie in Qj and let I be their intersection. If d lies in I then dc lies in all the Qj and so must be 0; hence d lies in Ann(c) = P; so I P. Consider some Qj which does not contain c; let Pj be the radical of Qj. If d lies in P = Ann(c) then dc = 0 lies in Qj and c does not lie in Qj so d must lie in Pj. In other words P is contained in the intersection jAPj. Since P is prime it contains the radical of I which is the intersection jAPj. Thus P must be equal to one of the primes Pj. We have shown that the associated primes are precisely the primes that occur as the radicals of the primary ideals in an irredundant primary decomposition.
1.5. Minimal primes. A prime ideal in R that is not contained in any other prime ideal is called a minimal prime of R. Let jQj be an irredundant primary decomposition of (0) in R. The radical of (0) consists of all nilpotents and is the thus the intersection of the radicals Pj’s of the Qj’s. Since the nil radical of a ring R is contained in all the prime ideals of R it follows that the minimal primes in R are the minimal elements in the collection of the Pj’s. The minimal prime ideals of a ring R are the minimal elements in the finite collection of associated primes of the ring R. The non-minimal associated primes are called embedded primes. Let Pk be an associated prime in R and let Q be the collection of all c such that there is some d not in P such that dc = 0. For any irredundant primary decomposition jQj, we see that Q is contained in Qk. The ideal Ik = jkQj is non-zero and its radical is jkPj. If Pk is a minimal prime then it does not contain the latter ideal and hence it does not contain Ik either. It follows that there is a d that lies in Ik but does not lie in Pk. For any c in Qk, the element dc lies in Ik Qk = (0) so is 0. Thus we see that Q = Qk. The primary ideal in an irredundant primary decompostion that corresponds to a minimal prime in R is uniquely determined.
1.6. Zero-divisors in a ring. From the description given above we see that the associated prime ideals consist of zero-divisors. Conversely, if xy = 0 and x does not lie in an associated prime Pj then y must lie in the corresponding Qj. So if x does not lie in any associated prime Pj then y lies in all the Qj’s and so must be 0. The collection of zero-divisors in R is thus precisely the union of all its associated primes.
1.7. Some geometry. We have seen that the primary decomposition is related to the decomposition of Spec(R) into irreducibles components. If we ignore for the moment, the distinction between irreducible components which differ only in their “nilpotent thickenings”, then the primary decomposition precisely corresponds to the decomposition into irreducible components. A zero-divisor c should be seen as a function that vanishes on some components but not others; the components on which it does not vanish are defined by the vanishing of those functions d such that dc is identically the zero function; that is, Ann(c). We have seen above that for any component, there is a function c such that Ann(c) is a primary ideal associated to that component; this is a function that vanishes on all components except the chosen one. This interprepretation makes sense unless the corresponding prime ideal is embedded (i. e. non-minimal). How can a function vanish on a component but apparently not on a closed subset? What can and does happen is that the function vanishes on the larger component but is nilpotent and non-zero on the embedded component which is contained in this larger component.
The primes associated with a component consist of functions that are nilpotent on that component. The above description also leads to characterisation of functions c that vanish on a non-embedded component as those for which there is function d which is not nilpotent on this component such that dc identically vanishes.
Finally, while dealing with primary decomposition, it is easy (but erroneous) to loose track of the elements of R which are not zero-divisors. We will see these elements acquire importance in a later section.