Characterizations of perfect recall

This paper considers the condition of perfect recall for the class of arbitrarily large discrete extensive form games. The known definitions of perfect recall are shown to be equivalent even beyond finite games. Further, a qualitatively new characterization in terms of choices is obtained. In particular, an extensive form game satisfies perfect recall if and only if the set of choices, viewed as sets of ultimate outcomes, fulfill the “Trivial Intersection” property, that is, any two choices with nonempty intersection are ordered by set inclusion.     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Dual automorphism-invariant modules over perfect rings

Under study are the dual automorphism-invariant modules and pseudoprojective modules. Some conditions were found under which the dual automorphism-invariant module over a perfect ring is quasiprojective. We also show that if R is a right perfect ring then a pseudoprojective right R-module M is finitely generated if and only if M is a Hopf module.     

On H-supplemented modules over a right perfect ring

In 1971, Koehler [11 Koehler, A. (1971). Quasi-projective and quasi-injective modules. Pac. J. Math. 36(3):713–720.[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu–Jans [22 Wu, L. E. T., Jans, J. P. (1967). On quasi-projectives. Illinois J. Math. 11:439–448. [Google Scholar]]. Later Mohamed–Singh [17 Mohamed, S. H., Singh, S. (1977). Generalizations of decomposition theorems known over perfect rings. J. Aust. Math. Soc. Ser. A 24(4):496–510.[Crossref] , [Google Scholar]] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18 Oshiro, K. (1983). Semiperfect modules and quasi-semiperfect modules. Osaka J. Math. 20:337–372.[Web of Science ®] , [Google Scholar]] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D₂) or (D₃) over right perfect rings.     

On the semi-perfect elimination

We give algorithmic characterizations of two classes of graphs, for which every ordering produced by the Lexicographic Breadth-First Search and the Maximum Cardinality Search, respectively, satisfies a prescribed property. These characterizations allow us to design linear-time optimization algorithms for these classes of graphs.     

A note on multiplicity of perfect modules of codimension one

In this note, we prove the following inequality for any perfect vl-module M of codimension one and for any maximal Buchsbaum A-module N of positive depth over a Cohen-Macaulay local ring A : e(M? N) ?μ(M).e(N)     

Regularly weakly based modules over right perfect rings and Dedekind domains

A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study

1. (1)
   rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and
    
2. (2)
   regularly weakly based modules over Dedekind domains.
    

     

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ?(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.     

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.     

Elementary equivalence of incidence rings over semi-perfect rings

In this paper, we prove that if two incidence rings constructed by the same semiperfect ring and some two quasi-ordered sets are elementarily equivalent, then the given sets are elementarily equivalent.     

On perfect and multiply perfect numbers

Summary Denote by P(x) the number of integers n≤x satisfying σ(n)≡0 (mod n), and by P ₂ (x) the number of integers n ≤ x satisfying σ(n)=2n. The author proves that P(x)<x ^3/4+ɛ and P ₂ (x)<x ^(1−c)/2 for a certain c>0.     

Crossed modules and doi-hopf modules

We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product. The second and the third author both thank the University of Brussels for its warm hospitality during their visit there. This author was supported partially by the CNCSU nr. 221 and by the Romanian academy, grant nr. 3721.     

Varieties of modules and affine modules

Acta Mathematica Hungarica -