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CONSTRUCTING (NON-)CATENARIAN RINGS
Y. AZIMI
Abstract. Let Rand Sbe commutative rings with identity, f:R→Sa ring
homomorphism and Jan ideal of S. Then the subring R ./fJ:= {(r, f(r)+j)|
r∈Rand j∈J}of R×Sis called the amalgamation of Rwith Salong Jwith
respect to f. In this paper, we consider the transfer of catenarity between R
and R ./fJ. As an application, we construct non-catenary rings and catenary
rings with special properties.
1. Introduction
D’Anna, Finocchiaro, and Fontana in [8] and [9] have introduced the following
construction. Let Rand Sbe two commutative rings with identity, let Jbe an ideal
of Sand let f:R→Sbe a ring homomorphism. They introduced the following
subring
R fJ:= {(r, f(r) + j)|r∈Rand j∈J}
of R×S, called the amalgamated algebra (or amalgamation) of Rwith Salong J
with respect to f. This construction generalizes the amalgamated duplication of
a ring along an ideal (introduced and studied in [11]). Moreover, several classical
constructions such as Nagata’s idealization (cf. [15, page 2], the R+X S[X] and the
R+XSJXKconstructions can be studied as particular cases of this new construction
(see [8, Example 2.5 and Remark 2.8]). The construction has proved its worth
providing numerous examples and counterexamples in commutative ring theory.
Until 1956, when Nagata constructed non-catenary local rings in [14], non-
catenary Noetherian rings considered not to exist. Since that time, there have been
many attempts to explore catenarity of rings with given properties. These efforts
lead to construction of non-catenary rings that are for example integrally closed
domains, UFDs, ... ([16], [12],...). Some of these efforts concerns with catenarity of
rings that are special cases of amalgamated algebra, e.g. [2], [3].
Recently, many properties of amalgamations investigated in several papers. In
particular, Cohen-Macaulay and (quasi-)Gorenstein properties investigated in [9],[17],
[4], and [5], and Pr¨ufer-like conditions in [13] and [6]. Note that Pr¨ufer domains and
Cohen-Macaulay rings are (universally) catenary. In this paper we pursue previous
works and investigate catenarity of amalgamations.
The outline of the paper is as follows. In Section 2, we state some essential
results on which we base our approach. More precisely, we explore when two prime
ideals of R fJare comparable. In Section 3, we classify some necessary and
sufficient conditions for the amalgamated algebra A fJto be catenary (Theo-
rem 3.2). Among the applications of our results are the classification of when the
2010 Mathematics Subject Classification. Primary 13C15, 13C14, 13A15, 13B99.
Key words and phrases. Amalgamated algebra, amalgamated duplication, Catenary ring, triv-
ial extension.
1
2 Y. AZIMI
amalgamated duplication of a ring along an ideal and the trivial extension of a ring
by a module are catenary (Corollary 3.3 and Corollary3.5). Our attempt results
in examples of the rings that are catenary but not Cohen-Macaulay or pr¨ufer. An
example of pr¨ufer ring (with zero-divisors) that is not catenary is given too. We
also provide a theorem (with a constructive proof) and some original examples of
a non-catenary rings. (Theorem 3.9, Example 3.10, and Example 3.11).
2. Comparability
To facilitate the reading of the paper, in this section, we recall notations and
definitions we need later. We also explore when a prime ideal of R fJis a (pure)
subset of the other one, and when such an inclusion is saturated.
Let us first fix some notation which we shall use throughout the paper: Rand
Sare two commutative rings with identity, Jis a non-zero proper ideal of the ring
S, and f:R→Sis a ring homomorphism. For a commutative ring A, the set
of nilpotent elements, prime ideals, and maximal ideals of Awill be denoted by
Nil(A), Spec(A), and Max(A) respectively. V(I) denotes the set of prime ideals of
Acontaining I. For a multiplicatively closed subset Tof A, we use the notation
T−1Ato denote the ring of fractions of Awith respect to T. For abbreviation, we
write “iff” instead of “if and only if”.
Remark 2.1. ([10, Corollary 2.5]) For p∈Spec(R) and q∈Spec(S)\V(J), set
p0f:=pfJ:= {(p, f(p) + j)|p∈p, j ∈J},
qf:={(r, f(r) + j)|r∈R, j ∈J, f(r) + j∈q}.
Then, the prime ideals of R fJare of the type p0for qf, for pin Spec(R) and
qvarying in Spec(S)\V(J). We call prime ideals of the form p0fas type 1 prime,
and prime ideals of the form qfas type 2 prime. Note that R fJdoes not have
any type 2 prime iff J⊆Nil(S). Note also that no type 1 prime is subset of a type
2 prime (since, for any j∈J, (0, j) is in any type 1 prime).
The key to the main results of this paper is given by the following elementary
lemmas on the comparability of prime ideals of R fJ.
Lemma 2.2. For i= 1,2, let pi∈Spec(R)and qi∈Spec(S)\V(J). Then
(1) p1⊆p2iff p0f
1⊆p0f
2
(2) p1(p2iff p0f
1(p0f
2
(3) q1⊆q2iff q1f⊆q2f
(4) q1(q2iff q1f(q2f
Proof. The proof of (1) and (2) follow immediately from definition of p0fin Remark
2.1. The proof of (3) and (4) are similar, and we give the proof only for one direction
of part (4). Assume that q1(q2, and pick y∈q2\q1. Let v∈J\q2. Hence,
yv ∈q2∩J\q1, Hence that (0, yv )∈q2\q1, as desired.
One can rather easily derive the following lemma as a consequence of the defini-
tions of p0fand qfin Remark 2.1.
Lemma 2.3. Let p∈Spec(R)and q∈Spec(S)\V(J). Then qf⊆p0fiff f−1(q+
J)⊆p. If moreover fis surjective, then qf⊆p0fiff f−1(q) + f−1(J)⊆p.
CONSTRUCTING (NON-)CATENARIAN RINGS 3
The following lemma, which is of independent interest, has important role in the
proof of Theorem 3.2.
Lemma 2.4. Let p∈Spec(R),q∈Spec(S)\V(J), and fbe surjective. Then
qf(p0fis saturated iff f−1(q)(pis saturated and f−1(J)⊆p.
Proof. Assume that qf(p0fis saturated. By Lemma 2.3, f−1(q) + f−1(J)⊆p.
To see that f−1(q)6=p, take f(a)∈J\q,a∈R. Hence (a, 0) ∈qf(p0f, hence
that a∈p\f−1(q). Next, on the contrary, suppose that f−1(q)(n(pbe a
chain of prime ideals of R. Two cases arise: If f−1(J)⊆n, then qf(n0f(p0f
by Remark 2.1, Lemma 2.2, and Lemma 2.3. If f−1(J)*n, then J*f(n) since
kerf ⊆n. By Lemma 2.2, qf(f(n)f. On the other hand, since kerf ⊆p, we
have f−1(f(n)) ⊆p, and so it follows from Lemma 2.3 that f(n)f(p0f. Hence the
inclusion qf(p0f, is not saturated, contradiction.
Assume conversely that f−1(q)(pis saturated and f−1(J)⊆p, and suppose
on the contrary that qf(p0fis not saturated, say qf(N(p0ffor some N ∈
Spec(R fJ). If N=nffor some n∈Spec(S)\V(J), then, by Lemma 2.3,
f−1(n)⊆p. But, as in the beginning of the proof, f−1(n)6=p. Therefore we have
f−1(q)(f−1(n)(p, contradiction. If N=n0ffor some n∈Spec(R),then, as in
the previous case, we have f−1(q)(n(p, contradiction.
3. Transfer of catenary condition in the amalgamations
In this section, we characterize catenarity of R fJwhen f:R→Sis surjec-
tive. Then we provide conditions under which R fJis not catenary. As a result,
we construct examples of non-catenary rings of arbitrary krull dimension and with
arbitrary difference in length between two saturated chains of prime ideals with the
same start and end points.
Let us first introduce the concept of catenary condition for an arbitrary subset
of Spec(R).
Definition 3.1. Let X⊆Spec(R). We call Xcatenary if, for each pair P⊆Q
of prime ideals in X, all saturated chains of prime ideals from Pto Qhave the
same finite length. Otherwise, we call it non-catenary. We simply call Rcatenary
if, Spec(R)is catenary.
Note that Ris catenary (in the usual sense), if it is catenary in the sense of
above definition. The following theorem is one of the main results of this paper.
Theorem 3.2. Let f:R→Sbe surjective. Then R fJis a catenary ring iff
both Rand Spec(S)\V(J)are catenary.
Proof. Assume that R fJis catenary. Then Ris catenary, since it is homomor-
phic image of R fJ. Next, by Lemma 2.2, any saturated chain of prime ideals in
Spec(S)\V(J) leads to a saturated chain of prime ideals in Spec(R fJ) of the
same length. Thus Spec(S)\V(J) is catenary.
Conversely, Assume that Rand Spec(S)\V(J) are catenary. Note that no type
1 prime can be a subset of type 2 prime. Thus any chain of prime ideals of R fJ
is in the form of one of the following cases:
Case 1. All elements of the chain are type 1 prime. Let p0f
0(· · · (p0f
nbe such
a saturated chain. By Lemma 2.2, this leads to the saturated chain p0(· · · (pn.
But, since Ris catenary, the length of any chain between p0and pnis n, and the
4 Y. AZIMI
desired result follows again from Lemma 2.2.
Case 2. All elements of the chain are type 2 prime. This case is similar to the
previous case, since Spec(S)\V(J) is catenary
Case 3. The chain starts with type 2 prime and ends with type 1 prime. On the
contrary, suppose that
q1f(· · · (qtf(p0f
1(· · · (p0f
s
qα1
f(· · · (qαt
f(p0f
α1(· · · (p0f
αs
be two saturated chains of different lengths in Spec(R fJ), with the same start
and end points and with t, αt, s, αs∈N. Then, by surjective assumption on f,
Lemma 2.2, and Lemma 2.4 we have saturated chains
f−1(q1)(· · · (f−1(qt)(p1(· · · (ps
f−1(qα1)(· · · (f−1(qαt)(pα1(· · · (pαs
of different lengths in Spec(R) (with the same start and end points) which contra-
dicts the catenarity of R.
In the above theorem, we assumed that fis surjective. In fact, if fis not
surjective, then the resulting statement is not always true. We shall illustrate this
with Example 3.10.
Recall that if f:= idRis the identity homomorphism on R, and Iis an ideal of
R, then R I := R idRIis called the amalgamated duplication of Ralong I.
Corollary 3.3. Let I be a non-zero ideal of R. Then R I is a catenary ring iff
so is R.
Proposition 3.4. Let J⊆Nil(S). Then R fJis catenary iff so is R.
Proof. By Remark 2.1, any chain of prime ideals of R fJconsists only of type
1 primes. Then, similar to the (case 1 in the) proof of Theorem 3.2, Lemma 2.2
completes the proof.
Let M(respectively, N= (Mi)n
i=1) be an R-module (respectively, a family of R-
modules). Then RnM(respectively, RnnN) denotes the trivial extension of Rby
M(respectively, the n-trivial extension of Rby N). It should be noted that both
construction are special cases of amalgamation with Jn= 0 (For definition and
more details see [8], [5]). Hence the next result follows from the above proposition.
Corollary 3.5. Let Mbe an R-module and N= (Mi)n
i=1 be a family of R-modules.
Then the following hold:
(1) RnnNis catenary iff so is R.
(2) RnMis catenary iff so is R.
As an application of our results we construct an example of a catenary ring
which is not Cohen-Macaulay. Recall from [4, Corollary 4.2] that if Ris local, Jis
contained in the Jacobson radical of Sand it is finitely generated as an R-module,
then R fJis Cohen-Macaulay iff Ris Cohen-Macaulay and Jis a maximal
Cohen-Macaulay R-module.
Example 3.6. Let kbe a field and x, y, z be algebraically independent indetermi-
nates over k. Let R0=k[x, y, z ], and T=R0\ hx, y, zi. Let R=T−1R0and
I=hx/1i. Then, by Corollary 3.3 and [4, Corollary 4.2],R I is a catenary ring
which is not Cohen-Macaulay, since Iis not a maximal Cohen-Macaulay R-module.
CONSTRUCTING (NON-)CATENARIAN RINGS 5
Pr¨ufer domains have several different characterizations, many of which have
been extended to the case of rings with zero-divisors. Among them it is commonly
accepted to define Pr¨ufer rings as the rings in which every non-zero finitely gener-
ated regular ideal is projective. It is well known that pr¨ufer domains are catenary.
But the next example shows that this is not true for Pr¨ufer rings.
Example 3.7. Let (R, m)be a local integral domain which is not catenary (see
for example [12, Theorem 10]). Then, by [7, Theorem 3.1(1)], the trivial extension
Rn(R/m)is a total ring of quotients, hence, a Pr¨ufer ring, while it is not catenary
by Corollary 3.5.
In the following, we give an example of catenary ring that is not pr¨ufer. To this
end, we appeal to [6, Corollary 3.12] which says if R I is a Pr¨ufer ring, then R
is a Pr¨ufer ring and Im=rImfor every m∈Max(R) and every r∈Reg(R).
Example 3.8. Let kbe a field and x, y, z be algebraically independent indetermi-
nates over k. Let R=S=k[x, y, z],f=idR, and I=hxi. Then, by Corollary
3.3 and [6, Corollary 3.12],R I is a catenary ring (of dimension 3) which is not
pr¨ufer. ([4, Example 4.12]).
We used the above results to construct new non-catenary (respectively, catenary)
rings from previously known non-catenary (respectively, catenary) rings. The fol-
lowing theorem provides a simple way for constructing non-catenary rings by means
of two catenary rings. this, together with its examples, is one of the main results
of the paper. We say that a chain p1(· · · (pksatisfying property Pis maximal
if no chain of the form p1(· · · (pk(pk+1 satisfies property P.
Theorem 3.9. Let pf−1(J)∈Max(R). Let n(p1(· · · (pkand n(q1(· · · (
ql, with k6=l, be two (saturated) maximal chains of elements of Spec(S)\V(J)
such that pk+J6=Sand qk+J6=S. Then R fJis not catenary.
Proof. Let pf−1(J) = m. Then pf−1(J+pk) = mand pf−1(J+ql) = m.
Hence Lemma 2.3 implies that pkf(m0fand qlf(m0f. Note that pkfand qlf
are not contained in any type 1 prime other than m0f, since pf−1(J) = m. On
the other hand, by the maximality of the above chains, one can not insert any
type 2 prime in the chains pkf(m0fand qlf(m0f. Therefore, by Lemma 2.2
and Lemma 2.3, we have the following saturated chain of prime ideals of different
lengths (between nfand m0f):
nf(p1f(· · · (pkf(m0f
nf(q1
f(· · · (qlf(m0f.
Example 3.10. Let R=k[z],S0=k[x1, . . . , xk, y1, . . . , yl, z],k < l, and T=S0\
hz, x1, . . . , xki∪hz, y1, . . . , yli. Let S=T−1S0,J=T−1(z), and f:R→Sbe the
natural ring homomorphism. For 1≤i≤kand 1≤j≤l, let pi=T−1hx1, . . . , xii,
and qj=T−1hy1, . . . , yji.
Then 0(p1(· · · (pkand 0(q1(· · · (qlare saturated chains of elements of
Spec(S)\V(J)of different lengths. Also, pf−1(J+pk) = hziand pf−1(J+ql) =
hzi. Thus, by Theorem 3.9, R fJis not catenary.
Note that the difference in length between the two saturated chains of prime ideals
from 0fto the maximal ideal hzi0fcan be made arbitrarily large; it is l−k.
6 Y. AZIMI
In the above example, Rand Sare both Cohen-Macaulay, while R fJis not
catenary. The situation would be more interesting; One can take R=ka field and
R fJwould be a domain. The following example, which is more illustrative,
deals with this case.
Example 3.11. Let R=k,S0=k[x, y1, y2, z],T=S0\ hz , xi ∪ hz, y1, y2i, and
S=T−1S0. Let J=T−1(z)and f:R→Sbe the natural ring homomorphism. Let
p=T−1hxi,q1=T−1hy1i,q2=T−1hy1, y2i,q3=T−1hy2i. Then, as in Example
3.10, R fJis not catenary. In the following, we list some properties of this ring:
(1) By [8, Proposition 5.2],R fJis an integral domain.
(2) Spec(S)\V(J) = {0,p,q1,q2,q3}. Thus, by Remark 2.1, Spec(R fJ) =
{0f,pf,q1f,q2f,q3f,00f}.
(3) dim R fJ= 3.
(4) 0f= 0 is the only minimal prime and 00fis the only maximal prime ideal
of R fJ.
We illustrate this example with Figure 1.
00f
q2f
pf
q1f
0f
Figure 1
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Department of Mathematics, University of Tabriz, Tabriz, Iran.
E-mail address:u.azimi@tabrizu.ac.ir
