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
Let R be a commutative ring with 1≠0.We introduce the concept of weakly 1-absorbing primary ideal,which is a generalization of 1-absorbing primary ideal.Aproperideal I of R is said tobeweakly1-absorbing primary if wh...Let R be a commutative ring with 1≠0.We introduce the concept of weakly 1-absorbing primary ideal,which is a generalization of 1-absorbing primary ideal.Aproperideal I of R is said tobeweakly1-absorbing primary if whenevernonunit elements a,b,c∈R and O≠abc∈I,we have ab∈I or c∈√I.A number of results concerning weakly 1-absorbing primary ideals are given,as well as examples of weakly 1-absorbing primary ideals.Furthermore,we give a corrected version of a result on 1-absorbing primary ideals of commutative rings.展开更多
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
We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such...We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.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Let R be a commutative ring with 1≠0.We introduce the concept of weakly 1-absorbing primary ideal,which is a generalization of 1-absorbing primary ideal.Aproperideal I of R is said tobeweakly1-absorbing primary if whenevernonunit elements a,b,c∈R and O≠abc∈I,we have ab∈I or c∈√I.A number of results concerning weakly 1-absorbing primary ideals are given,as well as examples of weakly 1-absorbing primary ideals.Furthermore,we give a corrected version of a result on 1-absorbing primary ideals of commutative rings.
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.
基金Acknowledgements the article and made The authors are thankful to numerous helpful suggestions the referees for their carefully reading This work was partially supported by the National Natural Science Foundation of China （Nos. 11071062, 11271208）, the Natural Science Foundation of Hunan Province （14JJ6027）, and the Scientific Research ~nd of Education Department of Hunan Province （10A033, 12C0130）.
文摘We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.
