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1.
Klaus Keimel 《Semigroup Forum》1971,2(1):55-63
Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone
semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e1 ε P and e2 ε Ann(P) such that x→(e1x, e2x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that
is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural
bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary
sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed.
An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico,
Mayaguez, Puerto Rico, March 9–13, 1970. 相似文献
2.
In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S. 相似文献
3.
Karen D. Aucoin 《Semigroup Forum》1996,52(1):157-162
A topological semigroupS is said to have thecongruence extension property (CEP) provided that for each closed subsemigroupT ofS and each closed congruence σ onT, σ can be extended to a closed congruence
onS. (That is,
∩(T xT=σ). The main result of this paper gives a characteriation of Γ-compact commutative archimedean semigroups with the congruence
extension property (CEP). Consideration of this result was motivated by the problem of characterizing compact commutative
semigroups with CEP as follows. It is well known that every commutative semigroup can be expressed as a semilattice of archimedean
components each of which contains at most one idempotemt. The components of a compact commutative semigroup need not be compact
(nor Γ-compact) as the congruence providing the decomposition is not necessarily closed. However, any component with CEP which
is Γ-compact is characterized by the afore-mentioned result. Characterization of components of a compact commutative semigroup
having CEP is a natural step towar characterization of the entire semigroup since CEP is a hereditary property. Other results
prevented in this paper give a characterization of compact monothetic semigroups with CEP and show that Rees quotients of
compact semigroups with CEP retain CEP. 相似文献
4.
set and denoted by (X;σ), and construct an inverse subsemigroup {\cal LI}_{(X;σ)}, say, of the symmetric inverse semigroup {\cal
I}_X. Next, we shall define a multiplication and a unary operation on {\cal LI}_{(X;σ)} by using the structure sandwich set {\cal M} of (X;σ), and make it a locally inverse *-semigroup which is denoted by {\cal LI}_{(X;σ)}({\cal M}). Finally, we
shall show that a locally inverse *-semigroup can be embedded in the ι-symmetric locally inverse *-semigroup {\cal LI}_{(X;σ)}({\cal M}) on an ι-set (X;σ). 相似文献
5.
V. D. Derech 《Ukrainian Mathematical Journal》2012,63(9):1390-1399
For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms
are permutable. 相似文献
6.
N. A. Vavilov 《Journal of Mathematical Sciences》2007,147(5):6995-7004
7.
Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases
where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where
the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata.
We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set. 相似文献
9.
David E. Dobbs 《Rendiconti del Circolo Matematico di Palermo》2009,58(3):327-336
If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R
P
/P R
P
and T
Q
/QT
Q
, where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation
domain R with σ(R)=∞. 相似文献
10.
S-内射模及S-内射包络 总被引:1,自引:0,他引:1
设R是环.设S是一个左R-模簇,E是左R-模.若对任何N∈S,有Ext_R~1(N,E)=0,则E称为S-内射模.本文证明了若S是Baer模簇,则关于S-内射模的Baer准则成立;若S是完备模簇,则每个模有S-内射包络;若对任何单模N,Ext_R~1(N,E)=0,则E称为极大性内射模;若R是交换环,且对任何挠模N,Ext_R~1(N,E)=0,则E称为正则性内射模.作为应用,证明了每个模有极大性内射包络.也证明了交换环R是SM环当且仅当T/R的正则性内射包e(T/R)是∑-正则性内射模,其中T=T(R)表示R的完全分式环,当且仅当每一GV-无挠的正则性内射模是∑-正则性内射模. 相似文献
11.
A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x2 = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of Kn if n + 1 = pq for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring. 相似文献
12.
In this paper we extend the construction of the field of rational numbers from the ring of integers to an arbitrary commutative ordered semigroup.We first construct a fractional ordered semigroup and a homomorphism ψs:R→S-1R.Secondly,we characterize the commutative ordered semigroup so constructed by a universal mapping property. 相似文献
13.
Prezemysław Kranz 《Semigroup Forum》1979,18(1):293-305
Let S be a commutative semigroup and So a subsemigroup. The present paper establishes a necessary and sufficient condition in order that any real additive functional
ϕ0 defined on So and dominated there by a real subadditive functional p defined on S, admit an additive extension to S such that ϕ≤p. This
result is a strengthening of a result of R.Kaufman [3]. From this easily follow recent results of Kobayashi [5] and Putcha,
Tamura [9] on extension of semigroup homomorphisms. Also a result of Ross on extension of semicharacters can be deduced from
this result.
As an application, the existence of continuous semicharacters on an open subsemigroup of an abelian topological group is derived.
This too, generalizes previous results (Gleason in [10]).
Supported in part by the Danish Natural Science Research Council 相似文献
14.
left order in Q and Q is a semigroup of left quotients of S if every q∈Q can be written as q=a^*b for some a, b∈S where a^* denotes the inverse of a in a subgroup of Q and if,
in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative
cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order
in Q then Q is completely regular and the {\cal D}-classes of Q are left groups. The semigroup S is right reversible and its
group of left quotients is the minimum semigroup of left quotients of S.
The authors are grateful to the ARC for its generous financial support. 相似文献
16.
We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions
of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B
σ
(X) in the case where σ(X) is a quasi-order. 相似文献
17.
Ernie Manes 《Semigroup Forum》2006,72(1):94-120
The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation
subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on
X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is
a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed
paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature
but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup
underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup
S contains a canonical subsemilattice g⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g⋆ ≅ L is obtained. 相似文献
18.
Mario Petrich 《Semigroup Forum》2005,71(3):337-365
On any regular semigroup S, the least group congruence σ, the greatest idempotent separating congruence μ and the least band
congruence β are used to give the T-classification of regular semigroups as follows. These congruences generate a sublattice
Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations
on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category T whose morphisms are surjective K-preserving homomorphisms subject to a mild condition. The class of regular semigroups is
made into a category T whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor
from T to T. The effect of the T-classification to P-semigroups is considered in some detail. 相似文献
19.
Richard Warlimont 《manuscripta mathematica》1992,77(1):361-383
Several classical results on multiplicative functions ℕ → ℂ are transposed to multiplicative functionsG → ℂ where (G, σ) denotes an additive arithmetical semigroup as introduced by John Knopfmacher. 相似文献
20.
P. M. Edwards 《Semigroup Forum》1989,39(1):257-262
For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective
semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the
maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ).
If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ)
or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ). 相似文献