Ascending Chain Condition Research Articles

Continuous Rings with Acc on Annihilators

AbstractIt is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

Finitely Embedded Commutative Rings

A theorem of Ginn and Moss [G-M] states that a right finitely embedded (= with finite essential right socle) two-sided Noetherian ring is Artinian. An example of Schelter and Small [S-S] can be applied to show that the theorem fails for right finitely embedded rings with the ascending chain conditions on right and left annihilators. We show here, however, that finitely embedded commutative rings with the acc on annihilator (= acc $\bot$) are Artinian. The proof uses the author’s characterization in [Fl] of acc $\bot$ rings, and the Levitzki [L] and Herstein-Small [H-S] theorem on the nilpotency of nil ideals in $2$-sided ace $\bot$ rings. A corollary is a result of Shizhong [Sh] that shows commutative subdirectly irreducible acc $\bot$ rings are QF.

Finitely embedded commutative rings

A theorem of Ginn and Moss [G-M] states that a right finitely embedded (= with finite essential right socle) two-sided Noetherian ring is Artinian. An example of Schelter and Small [S-S] can be applied to show that the theorem fails for right finitely embedded rings with the ascending chain conditions on right and left annihilators. We show here, however, that finitely embedded commutative rings with the acc on annihilator (= acc ⊥ \bot ) are Artinian. The proof uses the author’s characterization in [Fl] of acc ⊥ \bot rings, and the Levitzki [L] and Herstein-Small [H-S] theorem on the nilpotency of nil ideals in 2 2 -sided ace ⊥ \bot rings. A corollary is a result of Shizhong [Sh] that shows commutative subdirectly irreducible acc ⊥ \bot rings are QF.

Notes on $M$-semigroups

Let S be a torsion-free cancellativecommutative (additive) semigroup 2{0}. Let G be the quotient group of S. We assume G^S. For each subset A oi G, we set A~1={x^G: x+A<^S} and {A~1yl=Av. If AV=A for an ideal A of S, then A is called a v-ideal of S. If S satisfiesthe ascending chain condition for y-ideals,then S is called a Mori-semigroup. If S is a Mori-semigroup and and if each ideal of S generated by two elements is a v-ideal,then S is called an M-semigroup ([2]). If each ideal of S is a v-ideal, then S is called a reflexive semigroup. The maximal number n such that there exists a chain PiU^Pil^ ■・・Pn of prime ideals of S is denoted by dimS. If dimSS^l, then S has a unique maximal ideal. In this paper we study a semigroup version of a result ([1, Theoreme 3]) of Querre. Our result is the following.

The polynomial ring over a Goldie ring need not be a Goldie ring

Rings satisfying chain conditions have been of interest for quite some time. The famous Wedderburn Theorem states that semisimple Artinian rings must be finite direct sums of matrix rings. Goldie’s theorem links rings with ascending chain conditions to rings with descending chain conditions, thus extending Wedderburn’s result. A ring R is said to be a (right) Goldie ring if R satisfies the ascending chain condition on (right) annihilator ideals and R contains no infinite direct sum of (right) ideals. From Goldie’s theorem we know that a semiprime (right) Goldie ring must be an order in a semisimple (right) Artinian ring. A natural question to arise is whether the matrix and polynomial rings over (right) Goldie rings are also necessarily (right) Goldie rings, Under certain additional hypotheses such as if the ring is an order in a (right) Artinian ring [4] or if the ring contains a certain type of uncountable set in its center [I], the answer is yes. Moreover, the second Goldie condition is always preserved [3]. Hence, as in the case for the matrix ring counterexample [2], we must explore the ascending chain condition on (right) annihilator ideals. In the next section we construct a commutative Goldie ring R whose polynomial ring R[t] contains two infinite sets of polynomials, {p,(t)} and (qj(t)}, such that pi(t) qj(t) = 0 iff i #j. This condition forces qk(t) to be in Ann((p,(t): i>k)) and qk(t) to be excluded from Ann({p,(t): i>k1)). Thus R[t] has an infinite ascending chain of annihilator ideals

Continuous Rings with Acc on Essentials are Artinian

It is proved that a left continuous ring with ascending chain condition on essential left ideals is left artinian.

Continuous rings with ACC on essentials are Artinian

It is proved that a left continuous ring with ascending chain condition on essential left ideals is left artinian.

Correspondence theorems for modules and their endomorphism rings

Let RN be a left R-module and let B = End RM be its endomorphism ring. When investigating the relationship between properties of B and properties of M, one very useful and well-known technique makes use of two natural dual Galois connections, Gl and G2, which exist between the lattice, L, of submodules of M and the lattices, L, or L,, of right or left ideals of B. The Galois connection, Gl, is given by the maps rg: L -+ L, and i,: L, --f L, where rB( U) = {h f B: Uh = 0 >, for U G M, and I,(J) = {FIZZ M: mJ= 01, for JC B. Hcrc, the restrictions, YB and fJw, of rB and I, to the Galois objects of GI, E={UEL:U=~J here, the relevant ideals of B are all the finitely generated (f-g.) right ideals, since B is right noetherian if and only if it satisfies the ascending chain condition (a.c.c.) on f.g. right ideals. Now, when M is quasi-injective, every f.g. right ideal of B is a Galois object of G I. Moreover, FB and r, induce orderreversing bijections between the f.g. right ideals of B and the finitely closed submodules of M; hence, we can deduce that B is right noetherian if and only if A4 has the d.c.c. on finitely closed submodules [ 1, Corollary 4.3( 1 )I. 380 0021~8693,‘89 $3.00

QF-3 rings and torsion theories

AbstractIn this paper, the rings which have a torsion theory τ with associated torsion radical τ such that R/t(R) has a minimal τ-torsionfree cogenerator are studied. When τ is the trivial torsion theory these are precisely the left QF-3 rings. For τ = τL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on τL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators.

Ascending chain conditions and star operations

Ascending chain conditions and star operations

Monoids characterized by their quasi-injective s-systems

In this paper, S will denote a monoid, that is, a semigroup with identity i, which contains a zero 0. Each S-system is asumed to be right unitary (i.e. M1 = M) and centered (i.e. m0 = 0s = 0 for the zero 0 of M). A right S-system M S is injective if and only if, for every S-monomorphism g: Ps + Qs and for every Shomomorphism h: PS + MS' there exists an S-homomorphism h : QS + MS such that h g = h. Injective S-systems were originally introduced by Berthiaume [I], and later studied by many authors. Further references to this subject can be found in [4]. An S-system M S is quasi-injective if for N S ~ M S and every S-homomorphism f: N S + MS, there exists an S-homomorphism g: M S § M S such that giN = f" Every injective S-system is quasi-injective, but the converse is false. Quasiinjective S-systems have been studied by Lopez and Luedeman [6], and Satyanarayana [8]. Skornjakov ([10],Theorem 2) has proved that each direct sum of injective S-systems is injective if and only if S satisfies the ascending chain condition on right ideals. This result is not true if injectives are replaced by quasi-injectives. One object of this paper is to give a characterization of monoids whose class of quasi-injective S-systems is closed under the formation of direct sums. We shall also consider monoids all of whose S-systems are quasi-injectire. 2. PRELIMINARIES

The Existence of Minimal Regular Local Overrings for an Arbitrary Domain

It is shown that the set of regular local rings of dimension $n$ containing an integral domain $D$, having the same quotient field as $D$, and ordered by containment satisfies the descending chain condition. The set of regular local rings of dimension $n$ contained in a regular local ring of dimension $m,n > m$, need not satisfy the ascending chain condition, as is shown by example.

Some conditions for finiteness of a ring

Extending a result of Putcha and Yaqub, we prove that a non‐nil ring must be finite if it has both ascending chain condition and descending chain condition on non‐nil subrings. We also prove that a periodic ring with only finitely many non‐central zero divisors must be either finite or commutative.

On the class group of a Mori domain

It is well known that the monoid D(A) of the divisorial ideals of an integral domain A is a group if and only if A is completely integrally closed. In this case, one may also define the class group of A, C(A) = D(A)/P(A), where P(A) is the subgoup of principal ideals of A. The group C(A) sometimes gives good information about A; for example, if A is a Krull domain, C(A) = 0 if and only if A is a unique factorization domain [9, Proposition 6.11. In [6, 71, the class group C(A) is defined also for a noncompletely integrally closed domain A as C(A) = T(A)/P(A), where 7’(A) is the group of t-invertible t-ideals of A (see the definition in Sect. 1). The aim of this paper is to study the r-invertible t-ideals and the class group of a Mori domain. We recall that a Mori domain is a domain such that the ascending chain condition holds in the set of integral divisorial ideals. Noetherian domains are Mori domains and a Mori domain is a Krull domain if and only if it is completely integrally closed [9, Sect. 33. From this point of view, we begin by observing that in a Mori domain A the t-ideals are exactly the divisorial ideals (Proposition (1.1)); so that T(A) is the group of the invertible elements of D(A), i.e., the u-invertible ideals of A. Since in a Mori domain A a v-invertible divisorial prime is maximal divisorial (Proposition (1.3)), a complete characterization of u-invertible divisorial primes is found early on: they are the maximal divisorial primes P such that A, is a DVR (Corollary (1.4)). Also, we can give a characterization of a certain class of u-invertible divisorial ideals in terms of these primes, generalizing some well-known results for Krull domains

How far is a Mori domain from being a Krull domain?

A Mori domain is a domain such that the ascending chain condition holds in the set of integral divisorial ideals (cf. [13, p. 195]). Thus in a Mori domain A we can consider the proper integral ideals maximal with respect to being divisorial. We call them maximal divisorial ideals of A and we denote this set by DIn(A). The ideals of DIn(A) are primes (cf. Proposition 2.1) and have a central role in this paper. We first generalize to any Mori domain some results known for Krull domains, which are actually completely integrally closed Mori domains (cf. for example [4, Ch. 7, §1, No. 3, Theorem 2]). It is well known that, i rA is a Krull domain, Dm(A ) is the set of height 1 primes and A = ~ {Ap: PeDro(A)}, where this decomposition has finiteness character and, for each PeDro(A), Ap is a DVR (discrete valuation ring). Moreover in a Krull domain A the group of the divisorial ideals is generated by the height 1 primes ofA (cf. [4, Ch. 7, §1]). We show that, i fA is a Mori domain, A = ~ {Ap: PeDro(A)}, where this decomposition has finiteness character (cf. Proposition 2.2(b)) and it is irredundant (cf. Corollary 3.2), and we give a 'good representation' for any divisorial ideal of A in terms of DIn(A) (cf. Proposition 2.2(c)). In order to 'measure' the distance that a Mori domain A is from being Krull, it seems natural to divide Dm(A) into two subsets: the primes P of Dm(A ) such that A e is a DVR, and the other primes. We characterize the first subset of Dm(A) (denoted by ,~(A)) in Theorem 2.5. In this way we get for a Mori domain A a 'canonical decomposition' in two overrings, A =B CIA', where B = ~ {Ap: P e ~(A)} is a Krull domain and A'= ~ {Ap: PeDro(A) ~(A)} is a Mori domain very far from being Krull (we call it a strongly Mori domain) (cf. Theorem 3.3). Therefore,

Compact generation in partially ordered sets

AbstractSeveral “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.

The existence of minimal regular local overrings for an arbitrary domain

It is shown that the set of regular local rings of dimension n n containing an integral domain D D , having the same quotient field as D D , and ordered by containment satisfies the descending chain condition. The set of regular local rings of dimension n n contained in a regular local ring of dimension m , n &gt; m m,n &gt; m , need not satisfy the ascending chain condition, as is shown by example.

Almost Krull rings

AbstractThe notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

The ascending chain condition for real ideas

If A is a ring satisfying the ascending chain condition for real ideals, then this condition is also satisfied by the polynomial ring A[X]. However, an example is given to illustrate that the condition need not to hold in the power series ring A[[X]]. It is also shown that if every real prime ideal is the real ideal generated by finitely many elements, then the ring satisfies the ascending chain condition for real ideals. So, the analogues of Hilbert basis theorem and Cohen's theorem hold for real ideals.

Nil Derivations and Chain Conditions in Prime Rings

It is known that in a prime ring, a derivation nilpotent on a nonzero ideal must also be nilpotent in the entire ring. In this paper we show that a derivation of a prime ring is not necessarily nil even though it is nil on a nonzero ideal. It is nil if the ring satisfies the ascending chain condition on left (right) annihilator ideals.