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New results on C11 and C12 lattices with applications to Grothendieck categories and torsion theories
Toma ALBU,Mihai IOSIF
Front. Math. China. 2016, 11 (4): 815-828.
https://doi.org/10.1007/s11464-016-0550-y
In this paper, which is a cont inuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (Ci) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C11) or (C12). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.
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Valuation ideals and primary w-ideals
Gyu Whan CHANG,Hwankoo KIM
Front. Math. China. 2016, 11 (4): 829-844.
https://doi.org/10.1007/s11464-016-0554-7
Let D be an integral domain, V(D) (resp., t-V(D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and Nv= {f ∈ D[X] | c(f)v=D}. In this paper, we study integral domains D in which w-P(D) ⊆ t-V(D), t-V(D) ⊆ w-P(D), or t-V(D) = w-P(D). We also study the relationship between t-V(D) and V(D[X]Nv), and characterize when t-V(A + XB[X]) ⊆w-P(A + XB[X]) holds for a proper extension A ⊂ B of integral domains.
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On extensions of matrix rings with skew Hochschild 2-cocycles
Chan Yong HONG,Nam Kyun KIM,Tai Keun KWAK,Yang LEE
Front. Math. China. 2016, 11 (4): 869-900.
https://doi.org/10.1007/s11464-016-0552-9
We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.
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Koszul property of a class of graded algebras with nonpure resolutions
Jiafeng LÜ,Junling ZHENG
Front. Math. China. 2016, 11 (4): 985-1002.
https://doi.org/10.1007/s11464-016-0566-3
Given any integers a, b, c, and d with a>1, c≥0, b≥a + c, and d≥b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)- Koszul are provided.
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π-Armendariz rings relative to a monoid
Yao WANG,Meimei JIANG,Yanli REN
Front. Math. China. 2016, 11 (4): 1017-1036.
https://doi.org/10.1007/s11464-016-0561-8
Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a1g1+ a2g2+ · · · + angn, β = b1h1+ b2h2+ · · · + bmhm ∈ R[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.
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Unit groups of quotient rings of complex quadratic rings
Yangjiang WEI,Huadong SU,Gaohua TANG
Front. Math. China. 2016, 11 (4): 1037-1056.
https://doi.org/10.1007/s11464-016-0567-2
For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring Rdof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/〈vn〉 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/〈vn〉 for the cases d = −2,−3.
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17 articles
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