In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=（T 0 0 M U 0 N×UM N V）where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative ...In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=（T 0 0 M U 0 N×UM N V）where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative description of left Г-module as quintuple （A, B, C; f, g） with A ∈ mod T, B ∈ mod U and C ∈ mod V, f ： M ×T A →B ∈ mod U, g ： N ×U B → C ∈ mod V, we shall characterize uniform, hollow and finitely embedded modules over F, respectively. Also the radical as well as the socle of r （A ＋ B ＋ C） is determined.展开更多
It is shown that if M is an R-quasi-continuous left R-module and R satisfies ACC on left ideals of the form l(m), m ∈ M, then M is a direct sum of uniform sub-modules.
Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we ...Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.展开更多
基金the National Natural Science Foundation of China （No. 10371107）.
文摘In this paper we carry out a study of modules over a 3 × 3 formal triangular matrix ringГ=（T 0 0 M U 0 N×UM N V）where T, U, V are rings, M, N are U-T, V-U bimodules, respectively. Using the alternative description of left Г-module as quintuple （A, B, C; f, g） with A ∈ mod T, B ∈ mod U and C ∈ mod V, f ： M ×T A →B ∈ mod U, g ： N ×U B → C ∈ mod V, we shall characterize uniform, hollow and finitely embedded modules over F, respectively. Also the radical as well as the socle of r （A ＋ B ＋ C） is determined.
文摘It is shown that if M is an R-quasi-continuous left R-module and R satisfies ACC on left ideals of the form l(m), m ∈ M, then M is a direct sum of uniform sub-modules.
基金Supported by National Natural Science Foundation of China (Grant No. 10871016)
文摘Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.
