A Class of rings is said to be weakly hereditary if 0≠I△R∈ implies 0≠I^n∈ for some positive integer n, which generalizes the concept of heredity but other than that of regularity. In §2 the properties of su...A Class of rings is said to be weakly hereditary if 0≠I△R∈ implies 0≠I^n∈ for some positive integer n, which generalizes the concept of heredity but other than that of regularity. In §2 the properties of such class and its essential cover are studied. In §3 the upper radicals determined by them are investigated. At the same time the 'Problem 42, 44 and 55' of Szasz [7] are discussed. In §4 two examples are given, which show that the concept of weak heredity is independent of that of regularity.展开更多
For an arbitrary class of rings M, we have studied, in this paper, some necessary and sufficient conditions for ψM to be closed under homomorphic images or essential extensions, or to be a semisimple class or heredit...For an arbitrary class of rings M, we have studied, in this paper, some necessary and sufficient conditions for ψM to be closed under homomorphic images or essential extensions, or to be a semisimple class or hereditary class. The main results are: Theorem 4.1 For an arbitrary class of rings M, the following are equivalent: (1) M is a semisimple class; (2) ψM = ψuψM; (3) M~*=(uψM)~*; (4) M^(**)■M~*■(uψM)~*. Theorem 4.3 For an arbitray class of rings M, the follawing are equivalent: (1) ψM is the semisimple class of a hereditary radical; (2) ψM is an essentially closed semisimple class; (3) M~*=(uψM)~* and M~* is essentially hereditary; (4) M~*=(uψM)~* and uψM is essentially hereditary; (5)M~*=(uψM)~* and ψM is essentially closed.展开更多
In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple desce...In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.展开更多
Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general st...Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general strongly semisimple radicals and investigate some properties of them in normal classes of algebras. This extends some known studies on the theory of radicals of various algebraic strutures.展开更多
F.A.Szasz has put forward the open problem 55 in [1]: Let K be the class of all subdirectly irreducible rings, whose Jacobson radical is (0). Examine the upper radical determined by the class K. In this paper, the pro...F.A.Szasz has put forward the open problem 55 in [1]: Let K be the class of all subdirectly irreducible rings, whose Jacobson radical is (0). Examine the upper radical determined by the class K. In this paper, the problem has been examined. (1) It has been proved that the upper radical R determined by the class K is a special radical,which lies between Jacobson radical and Brown-McCoy radical. (2) It has been given some necessary and sufficient condition of ring A to be an R-radical ring.展开更多
文摘A Class of rings is said to be weakly hereditary if 0≠I△R∈ implies 0≠I^n∈ for some positive integer n, which generalizes the concept of heredity but other than that of regularity. In §2 the properties of such class and its essential cover are studied. In §3 the upper radicals determined by them are investigated. At the same time the 'Problem 42, 44 and 55' of Szasz [7] are discussed. In §4 two examples are given, which show that the concept of weak heredity is independent of that of regularity.
文摘For an arbitrary class of rings M, we have studied, in this paper, some necessary and sufficient conditions for ψM to be closed under homomorphic images or essential extensions, or to be a semisimple class or hereditary class. The main results are: Theorem 4.1 For an arbitrary class of rings M, the following are equivalent: (1) M is a semisimple class; (2) ψM = ψuψM; (3) M~*=(uψM)~*; (4) M^(**)■M~*■(uψM)~*. Theorem 4.3 For an arbitray class of rings M, the follawing are equivalent: (1) ψM is the semisimple class of a hereditary radical; (2) ψM is an essentially closed semisimple class; (3) M~*=(uψM)~* and M~* is essentially hereditary; (4) M~*=(uψM)~* and uψM is essentially hereditary; (5)M~*=(uψM)~* and ψM is essentially closed.
文摘In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.
文摘Puczylowski established the general theory of radicals of the class of objects called algebras. In this paper, we make use of the method of lattice theory to characterize the general hereditary radicals and general strongly semisimple radicals and investigate some properties of them in normal classes of algebras. This extends some known studies on the theory of radicals of various algebraic strutures.
文摘F.A.Szasz has put forward the open problem 55 in [1]: Let K be the class of all subdirectly irreducible rings, whose Jacobson radical is (0). Examine the upper radical determined by the class K. In this paper, the problem has been examined. (1) It has been proved that the upper radical R determined by the class K is a special radical,which lies between Jacobson radical and Brown-McCoy radical. (2) It has been given some necessary and sufficient condition of ring A to be an R-radical ring.
