Let R be a commutative ring with nonzero identity and n be a positive integer.In this paper,we introduce and investigate a new subclass ofϕ-n-absorbing primary ideals,which are calledϕ-(n,N)-ideals.Letϕ:I(R)→I(R)∪{∅...Let R be a commutative ring with nonzero identity and n be a positive integer.In this paper,we introduce and investigate a new subclass ofϕ-n-absorbing primary ideals,which are calledϕ-(n,N)-ideals.Letϕ:I(R)→I(R)∪{∅}be a function,where I(R)denotes the set of all ideals of R.A proper ideal I of R is called aϕ-(n,N)-ideal if x1⋯xn+1∈I\ϕ(R)and x1⋯xn∉I imply that the product of xn+1 with(n−1)of x1,…,xn is in 0–√for all x1,…,xn+1∈R.In addition to giving many properties ofϕ-(n,N)-ideals,we also use the concept ofϕ-(n,N)-ideals to characterize rings that have only finitely many minimal prime ideals.展开更多
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
文摘Let R be a commutative ring with nonzero identity and n be a positive integer.In this paper,we introduce and investigate a new subclass ofϕ-n-absorbing primary ideals,which are calledϕ-(n,N)-ideals.Letϕ:I(R)→I(R)∪{∅}be a function,where I(R)denotes the set of all ideals of R.A proper ideal I of R is called aϕ-(n,N)-ideal if x1⋯xn+1∈I\ϕ(R)and x1⋯xn∉I imply that the product of xn+1 with(n−1)of x1,…,xn is in 0–√for all x1,…,xn+1∈R.In addition to giving many properties ofϕ-(n,N)-ideals,we also use the concept ofϕ-(n,N)-ideals to characterize rings that have only finitely many minimal prime ideals.
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.
