Let R be a commutative ring with 1≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ∈ R and a b ∈ I, we have a ∈ √I or b ∈√I; and I is a weakly semiprimary ideal of R if whenever a, b ∈ R...Let R be a commutative ring with 1≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ∈ R and a b ∈ I, we have a ∈ √I or b ∈√I; and I is a weakly semiprimary ideal of R if whenever a, b ∈ R and 0 ≠ ab ∈ √I, we have a ∈√I or b ∈ √I. In this paper, we introduce a new class of ideals that is closely related to the class of （weakly） semiprimary ideals. Let I（R） be the set of all ideals of R and let δ ： I（R） → I（R） be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J I, we have L δ（L） and δ（J） δ（I）. Let δ be an expansion function of ideals of R. Then a proper ideal I of R is called a δ-semiprimary （weakly δ-semiprimary） ideal of R if ab ∈ I （0 ≠ ab ∈ I） implies a ∈ δ（I） or b∈ δ（I）. A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given.展开更多
Is a semiprimary right self-injective ring a quasi-Frobenius ring？ Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing...Is a semiprimary right self-injective ring a quasi-Frobenius ring？ Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF： a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.展开更多
文摘Let R be a commutative ring with 1≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ∈ R and a b ∈ I, we have a ∈ √I or b ∈√I; and I is a weakly semiprimary ideal of R if whenever a, b ∈ R and 0 ≠ ab ∈ √I, we have a ∈√I or b ∈ √I. In this paper, we introduce a new class of ideals that is closely related to the class of （weakly） semiprimary ideals. Let I（R） be the set of all ideals of R and let δ ： I（R） → I（R） be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J I, we have L δ（L） and δ（J） δ（I）. Let δ be an expansion function of ideals of R. Then a proper ideal I of R is called a δ-semiprimary （weakly δ-semiprimary） ideal of R if ab ∈ I （0 ≠ ab ∈ I） implies a ∈ δ（I） or b∈ δ（I）. A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given.
文摘Is a semiprimary right self-injective ring a quasi-Frobenius ring？ Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF： a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.
