Special systems of ideals in commutative rings

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXV (1985)

A n d r z e j N o w ic k i (Torun) and R y s z a r d Z u c h o w s k i (Bydgoszcz)

Special systems of ideals in commutative rings

Introduction. Throughout this paper, all rings are commutative with identity, and all ring homomorphisms preserve the identity. Let R be a ring;

we denote by I(R) the set of all ideals in R.

If M cz I {R) satisfies conditions A1-A8 (see [10]), then by (R , M) we denote a system of ideals. Definitions and notions used here, were introduced in paper [10]. This paper contains three parts. In the first part, we character­

ize special systems, i.e., systems M for which A(R, M) — B (R, M). In the second part and third part we study topological properties of the space A(R, M) for any system (R, M) and next we prove some properties of A(R, M) if (R, M) is a special system.

Generalizing W. F. Keigher results ([5]), we construct retraction

q r : Spec R -» A( R, M) (see Theorem 3.1). We also present another proof of the fact that A( R, M) is a spectral space (cf. [8]).

1. Characterization of special systems.

D e f in it io n 1.1. A system (R, M ) satisfying the condition A( R, M)

= B(R, M) will be called a special system. Special systems form a sub­

category of the category of all systems.

T h e o r e m 1.2. Let (R, M) be a system. Following conditions are equivalent:

(1) (R, M ) is special.

(2) The radical of an arbitrary M-ideal is an M-ideal.

(3) Every primitive M-ideal is radical.

(4) I f P is a prime ideal in R, then P # is a prime ideal too.

(5) I f A e M and P is a minimal prime ideal containing A, then P e M . Proof is analogous to the proof of the similar theorem for differential systems. Equivalence of (2) and (4) see [5], (1) and (3) see [7].

E x a m p l e s . A system of homogeneous ideals, in a ring with a grading is

special. Similarly, a system of differential ideals with respect to a higher

derivation is special. A ring R with a derivation d for which the system of

differential ideals is special, we call d-MP ring ([3], [6], [7]) or a special differential ring ([5]).

T h e o r e m 1.3. Let (R , M) be a system with a Noetherian ring R.

Following conditions are equivalent.

(1) (R , M) is special.

(2) A ss^ /l) ci M for every M-ideal A.

P ro o f. (1)=>(2). Let A e M. By Theorem 3.4 [10] we have an irredund- ant primary decomposition A = Qx n Q2 n ... n Qn, with primary ideals Q i , . . . , Q n, which belong to M. Since Ass(A) = [r(Qx), ..., r(Qn)}, and because r(Qj), ..., r(Q„) belong to M, so Ass(^) <= M.

(2)=>(1). In the set AssK(.4) there are all minimal prime ideals containing A. Using (2) and Theorem 1.2 (5) we obtain the result.

T h e o r e m 1.4. Let (R, M) be a special system, where R is a Noetherian ring and let A be an ideal different from R. There exist М -prime ideals P x, P2, ..., Pn such that for every i = 1, 2, ..., n we have P, A zd P x

x P 2- .. .P „ .

P ro o f. Notice at first that Theorem is satisfied for primary M-ideals.

Suppose now that the family I of all M-ideals for which Theorem is not satisfied is non-empty. An ideal A maximal among elements of I is not M- irreducible ([10]), a remaining part of the proof is standard.

C o r o l l a r y 1.5. In every special system (R , M) with a Noetherian ring R zero ideal 0 is a product of a finite number of M-prime ideals.

2. A(R, M) as a topological space. The Zariski topology on Spec R induces on A(R, M) the topology in which closed sets are of the form V a (E) — [ P e A ( R , M): Р э £ ] , and open sets are of the form DA(E)

= { P e A ( R , M): P f i E ) , where E is an arbitrary subset of R.

D e f in it io n 2.1. If E cz R, then rA(E ) = f ] (P e A (P , M): P => E) will be called the A-radical of E. If Y <= A(R, M), then we write: J A( Y ) — f) J a (0) = R.

We can prove the following properties of rA, VA, J A.

T h e o r e m 2.2. Let E, F <= R, Y c: A( R, M). Then:

(1) rA(E-F) = rA( E ) n r A(F).

(2) V a (E) = М И ) = ^ (/■[£]) = VA(rA(E)).

(3) J a (Y) is a radical M-ideal.

(4) J A(VA(E)) = rA(E).

(5) VA{J a (Y)) = Y, where Y denotes the closure of Y in A(R, M).

(6) The mappings J A, VA are inverse, order reversing bijections, between the set of closed subsets of A(R, M ) and the set of radical M-ideals.

(7) There is a bisection between the family of non-empty closed irreducible

Systems o f ideals and commutative rings 111

sets in A(R, M), and the set of prime M-ideals. Precisely, if P e A ( R , M), then VA(P) is an irreducible closed set in A(R, M), and every irreducible closed set in A(R, M) is of this form.

A morphism of systems /: {R, M) ->(S, N ) induces the map af : Spec S -►Spec R, which further induces the continuous map Af : A(S, N) -> A ( R , M).

In this way, we obtain the contravariant functor from the category of systems, to the category of topological spaces.

T heorem 2.3. Let f: (R, M )->(S, N) be a morphism of systems. Then (1) if f: (R, M) -*(S, N ) is a morphism of systems, where f is a surjec­

tion, then Af : A(S, N) -►Л (JR, M) is a homeomorphism onto the closed subset VA(ker/ ) cz A(R, M).

(2) \m A f is a dense subset in A(R, M) if and only if / -1 (^(0)) <= ^ (0 ).

3. A(R, M ) as a spectral space. Let (R, M) be a special system. By Theorem 2.3, [10], we obtain that if P eS pec R, then P # e A ( R , M). Let

qr : Spec R -> A(R, M) be a map defined by qr {P) — P#.

T heorem 3.1. I f (R, M) is a special system, then

is an open retraction.

P ro o f. (1) If P e A ( R , M ) , then qr (P) = P # = P . Hence

= l/UR.M) •

(2) qr is a continuous map, because it satisfies a condition qr 1 (DA (E))

(3)

is an open map, because qr (P(E)) = DA(E).

L emma 3.2. I f f: (R, M) —► (S, N) is a morphism of special systems, then the diagram:

Spec S — Q/ Spec R

Ps PR

is commutative.

■

T heorem 3.3. Let (R, M) be a special system. Then:

(1) Open basis sets in A( R, M) of the form DA(x), are quasi-compact. In particular, A( R, M) is a quasi-compact space.

(2) A( R, M) is a dense subset of Spec R.

(3) An open set DA(E) in A (R, M) is quasi-compact if and only if DA(E)

= D a {A), where A is an M-ideal having an M-basis.

P ro o f. Notice that all open sets DA(x), where x e R , form a basis in A{R, M), because DA(E) = U DA(x).

x e E

(1) Since D a ( x ) — qr (D( x )) and D(x) is a quasi-compact set in Spec R, we have that DA(x) is a quasi-compact set as a continuous image of a quasi­

compact set.

(2) Because (R , M) is a special system so r^(0) = r(0). Hence A ( R , M) = K (f| {P'- P e A ( R , Щ ) = 7(^(0)) = 7(r(0)) = Spec R.

(3) Let D a (E) be a quasi-compact open set in A(R, M). Since DA(E)

= U ^D

⁽

^{) , so}

x eE

D a (£) = D a (X j ) u . . . u D a ( x „) = D a ({Xi, ..., x„}) = D a ([Xj, ..., x j) . A converse implication is trivial.

Recall the following definitions introduced by Hochster ([4]).

D e f in it io n 3.4. A topological space X is called spectral if:

(1) X is a T0-space.

(2) X is quasi-compact.

(3) Quasi-compact open sets are closed under finite intersection.

(4) Quasi-compact open subsets form an open basis.

(5) Every non-empty irreducible closed subset has a generic point.

D e f in it io n 3.5. A continuous map of spectral spaces is called spectral if the inverse image of quasi-compact sets are quasi-compact.

T h e o r e m 3.6. 1° I f (R, M) is a special system, then A(R, M) is a spectral space.

2° I f f ( R , M ) - + ( S , N ) is a morphism of special systems, then Af : ^4 (<S, N ) -> A(R, M) is a spectral map.

P ro o f. We verify conditions (l)-(5) of Definition 3.4.

(1) A subspace of T0-space if T0 too.

Conditions (2), (4) follow from Theorem 3.3.

(3) Let D a {A1), ..., D a (A„) be quasi-compact open sets in A( R, M). By Theorem 3.3 (3) it follows that A x = [B ,], ..., A„ = [B„], where B x, ..., Bn are finitely generated ideals of R. Therefore

DA( A l) n . . . n D A(A„) = DA(Bl ) n . . . n D A(B„)

— D a {B x ■ B 2 . . . B n) = D a ([B j . . . B„]), i.e., D a (Ax) n . . . n D A(Л„) is a quasi-compact open set.

(5) Let У be a non-empty irreducible closed set in A(R, M). Then Y = VA{P), where P e A ( R , M) (Theorem 2.3 (5)), so У = [P].

2° Let /: (R, M) -»(S, N ) be a morphism of special systems, and let U be a quasi-compact open set in A(R, M). Then U = DA([xx, ..., x„])

= ^ ( x i , ..., x„), where x x, x 2, x neR. So we have A j ' i U ) = ^ ^ ^ ( x j , ..., x„))

= DA( f M , . . . . / M ^ ^ f t / C x j ) , . . . , f ( x n)%

i.e., A J l {U) is a quasi-compact set in A(R, M).

Systems o f ideals and commutative rings 113

Using results of Hochster paper ([4]) we get

C o r o l l a r y 3.7. I f (R , M) is a special system, then there exists a ring R' such that spaces Spec R' and A( R, M) are homeomorphic.

C o r o l l a r y 3.8. I f /: (R, M) -> (S, N ) is a morphism of special systems, then there exist rings R', S' and a ring homomorphism h: R' -+ S', and there exist homeomorphisms a: A(R, M )-> Spec R' and ft: A(S, JV)->Spec S' such that the diagram:

A I S . N ) ----

a fi

Qh V

Spec S ' ---► Spec R '

is commutative.

E x a m p l e 1. Let R = K [ x ] be a polynomial ring over a field К of characteristic zero, and let d: R - + R be a non-zero derivation such that d(K) = 0. Moreover, let M be the set of all differential ideals of R.

If d(x) = f i 1---fnn, where / i , / 2, ...,/„ are pairwise different irreducible polynomials and alt ..., an non-negative integers, then A( R, M) is homeo­

morphic with S pecS - 1 Z, where S = Z \((p !)u ...u(p„)) and px, ...,p„ are different prime numbers.

E x a m p l e 2. Let JR = K [Xi, ..., x„] be a polynomial ring over a field К of characteristic zero and let d: R -> R be a non-zero derivation, such that d(K) = 0, d{Xi)eK, for i = 1, 2, ..., n. If M is the set of all differential ideals of R, then A (R, M) is homeomorphic with Spec K [ x 1? x 2, ...,

E x a m p l e 3. Let R — Z, M = {0, (p)", n = 0 ,1 , 2, ...}, where p is a fixed prime number. Then ^4(JR, M) is homeomorphic with Spec Z (p).

R e m a rk . Theorems analogous to Theorem 3.6 for the space A( R, M) can be proved also for the spaces B(R, M) and C{R, M) with topologies similar to the Zariski topology on Spec R but under an additional assumption that the ascending chain condition on M-ideals A such that rc (A) = A is satisfied, where rc (A) is an intersection of M-prime ideals containing A.

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~ Roczniki PTM — Prace Matematyczne XXV

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INSTITUTE O F MATHEMATICS, NICHOLAS COPERNICUS UNIVERSITY, TORUN, POLAND

and

INSTITUTE O F MATHEMATICS,

PEDAGOGICAL COLLEGE, BYDGOSZCZ, POLAND