In this paper it is shown that a sortable ideal I is Freiman if and only if its sorted graph is c... more In this paper it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.
We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determin... more We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals HS j (I) as functions of j, when I is an equigenerated squarefree Borel ideal.
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of so... more By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.
Let R be a commutative ring, π be a finite group, Rπ be the group ring of π over R. Theorem 1. If... more Let R be a commutative ring, π be a finite group, Rπ be the group ring of π over R. Theorem 1. If R is a commutative artinian ring and π is a finite group. Then the Cartan map c : K 0 (Rπ) → G 0 (Rπ) is injective. Theorem 2. Suppose that R is a Dedekind domain with char R = p > 0 and π is a p-group. Then every finitely generated projective Rπ-module is isomorphic to F ⊕ A where F is a free module and A is a projective ideal of Rπ. Moreover, R is a principal ideal domain if and only if every finitely generated projective Rπ-module is isomorphic to a free module. Theorem 3. Let R be a commutative noetherian ring with total quotient ring K, A be an R-algebra which is a finitely generated R-projective module. Suppose that I is an ideal of R such that R/I is artinian. Let {M 1 , . . . , M n } be the set of all maximal ideals of R containing I. Assume that the Cartan map c i : K 0 (A/M i A) → G 0 (A/M i A) is injective for all 1 ≤ i ≤ n. If P and Q are finitely generated A-projective modules with KP ≃ KQ, then P/IP ≃ Q/IQ.
We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associ... more We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n -cyclic graph, given some precise formulas for the depth when n ̸ ≡ 1 (mod 4) , tight bounds when n ≡ 1 (mod 4) , and for Stanley depth when n ≡ 0, 3 (mod 4) , tight bounds when n ≡ 1, 2 (mod 4) . We also give some formulas for the depth and Stanley depth of a quotient of the path ideals of length n -1 and n .
In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is ... more In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.
We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determin... more We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals HS j (I) as functions of j, when I is an equigenerated squarefree Borel ideal.
In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an a... more In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for reg(I t) for t ≥ 2. The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
We give some precise formulas for the depth of the quotient rings of the edge ideals associated t... more We give some precise formulas for the depth of the quotient rings of the edge ideals associated to a graph consisting, either of the union of some line graphs L 3r 1 ,. .. , L 3r k 1 , L 3s 1 +1 ,. .. , L 3s k 2 +1 and L 3t 1 +2 ,. .. , L 3t k 3 +2 or of the union of cycle graphs C 3r 1 ,. .. , C 3r k 1 , C 3s 1 +1 ,. .. , C 3s k 2 +1 and C 3t 1 +2 ,. .. , C 3t k 3 +2 , with a common vertex. We also give some tight bounds for their Stanley depths. Mathematics Subject Classification (2010): 13C15; 13P10; 13F20.
In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is ... more In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.
We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associ... more We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n-cyclic graph, given some precise formulas for depth when n ≡ 1 (mod 4), tight bounds when n ≡ 1 (mod 4) and for Stanley depth when n ≡ 0, 3 (mod 4), tight bounds when n ≡ 1, 2 (mod 4). Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length n -1 and n.
Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R... more Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $\pi$ is a finite group. Then the Cartan map $c:K_0(R\pi)\to G_0(R\pi)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $\pi$ is a $p$-group. Then every finitely generated projective $R\pi$-module is isomorphic to $F \oplus cA$ where $F$ is a free module and $cA$ is a projective ideal of $R\pi$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $R\pi$-module is isomorphic to a free module. Theorem 3. Let $R$ be a commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra which is a finitely generated $R$-projective module. Suppose that $I$ is an ideal of $R$ such that $R/I$ is artinian. Let $\{cM_1,\ldots,cM_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/cM_iA)\to G_0(A/cM_i...
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals I ⊂ K[x, y] o... more We consider the fiber cone of monomial ideals. It is shown that for monomial ideals I ⊂ K[x, y] of height 2, generated by 3 elements, the fiber cone F (I) of I is a hypersurface ring, and that F (I) has positive depth for interesting classes of height 2 monomial ideals I ⊂ K[x, y], which are generated by 4 elements. For these classes of ideals we also show that F (I) is Cohen-Macaulay if and only if the defining ideal J of F (I) is generated by at most 3 elements. In all the cases a minimal set of generators of J is determined.
To a simplicial complex ∆ , we associate a square-free monomial ideal F (∆) in the polynomial rin... more To a simplicial complex ∆ , we associate a square-free monomial ideal F (∆) in the polynomial ring generated by its facet over a field. Furthermore, we could consider F(∆) as the Stanley-Reisner ideal of another simplicial complex δN (F(∆)) from facet ideal theory and Stanley-Reisner theory. In this paper, we determine what families of simplicial complexes ∆ have the property that their Stanley-Reisner complexes δN (F(∆)) are shellable. Furthermore, we show that the simplicial complex with the free vertex property is sequentially Cohen-Macaulay. This result gives a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.
The aim of this study was to compare quantitative polymerase chain reaction (qPCR) with immunohis... more The aim of this study was to compare quantitative polymerase chain reaction (qPCR) with immunohistochemistry (IHC) for the detection of Her-2 in gastric cancer, and to investigate the correlation between the expression levels of human epidermal growth factor receptor 2 (Her-2) and clinical features. Clinical data from 426 cases of gastric cancer were collected. Her-2 expression levels in cancerous tissue were detected using IHC, and the Her-2/neu gene expression levels were determined by qPCR. The correlation between the expression level of Her-2 and clinical features was investigated. The positive expression rate of Her-2 in cancerous tissue detected using qPCR and IHC was 11.17% (46/412) and 13.38% (57/426), respectively. The positive expression of the Her-2 protein/gene was significantly correlated with the depth of invasion and lymphatic metastasis, as well as the TNM stage (P<0.05). No significant correlation was identified between positive expression of the Her-2 protein/ge...
Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal with almost mi... more Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal with almost minimal mixed multiplicity such that depth G(I) d − 1. We show that F m (I) has almost maximal depth (i.e. depth F m (I) d − 1).
In this paper it is shown that a sortable ideal I is Freiman if and only if its sorted graph is c... more In this paper it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.
We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determin... more We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals HS j (I) as functions of j, when I is an equigenerated squarefree Borel ideal.
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of so... more By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.
Let R be a commutative ring, π be a finite group, Rπ be the group ring of π over R. Theorem 1. If... more Let R be a commutative ring, π be a finite group, Rπ be the group ring of π over R. Theorem 1. If R is a commutative artinian ring and π is a finite group. Then the Cartan map c : K 0 (Rπ) → G 0 (Rπ) is injective. Theorem 2. Suppose that R is a Dedekind domain with char R = p > 0 and π is a p-group. Then every finitely generated projective Rπ-module is isomorphic to F ⊕ A where F is a free module and A is a projective ideal of Rπ. Moreover, R is a principal ideal domain if and only if every finitely generated projective Rπ-module is isomorphic to a free module. Theorem 3. Let R be a commutative noetherian ring with total quotient ring K, A be an R-algebra which is a finitely generated R-projective module. Suppose that I is an ideal of R such that R/I is artinian. Let {M 1 , . . . , M n } be the set of all maximal ideals of R containing I. Assume that the Cartan map c i : K 0 (A/M i A) → G 0 (A/M i A) is injective for all 1 ≤ i ≤ n. If P and Q are finitely generated A-projective modules with KP ≃ KQ, then P/IP ≃ Q/IQ.
We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associ... more We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n -cyclic graph, given some precise formulas for the depth when n ̸ ≡ 1 (mod 4) , tight bounds when n ≡ 1 (mod 4) , and for Stanley depth when n ≡ 0, 3 (mod 4) , tight bounds when n ≡ 1, 2 (mod 4) . We also give some formulas for the depth and Stanley depth of a quotient of the path ideals of length n -1 and n .
In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is ... more In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.
We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determin... more We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals HS j (I) as functions of j, when I is an equigenerated squarefree Borel ideal.
In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an a... more In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for reg(I t) for t ≥ 2. The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, ... more Let S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number sat(I) of Veronese type ideals I. Using this formula, we show that sat(I k) is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
We give some precise formulas for the depth of the quotient rings of the edge ideals associated t... more We give some precise formulas for the depth of the quotient rings of the edge ideals associated to a graph consisting, either of the union of some line graphs L 3r 1 ,. .. , L 3r k 1 , L 3s 1 +1 ,. .. , L 3s k 2 +1 and L 3t 1 +2 ,. .. , L 3t k 3 +2 or of the union of cycle graphs C 3r 1 ,. .. , C 3r k 1 , C 3s 1 +1 ,. .. , C 3s k 2 +1 and C 3t 1 +2 ,. .. , C 3t k 3 +2 , with a common vertex. We also give some tight bounds for their Stanley depths. Mathematics Subject Classification (2010): 13C15; 13P10; 13F20.
In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is ... more In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.
We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associ... more We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n-cyclic graph, given some precise formulas for depth when n ≡ 1 (mod 4), tight bounds when n ≡ 1 (mod 4) and for Stanley depth when n ≡ 0, 3 (mod 4), tight bounds when n ≡ 1, 2 (mod 4). Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length n -1 and n.
Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R... more Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $\pi$ is a finite group. Then the Cartan map $c:K_0(R\pi)\to G_0(R\pi)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $\pi$ is a $p$-group. Then every finitely generated projective $R\pi$-module is isomorphic to $F \oplus cA$ where $F$ is a free module and $cA$ is a projective ideal of $R\pi$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $R\pi$-module is isomorphic to a free module. Theorem 3. Let $R$ be a commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra which is a finitely generated $R$-projective module. Suppose that $I$ is an ideal of $R$ such that $R/I$ is artinian. Let $\{cM_1,\ldots,cM_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/cM_iA)\to G_0(A/cM_i...
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals I ⊂ K[x, y] o... more We consider the fiber cone of monomial ideals. It is shown that for monomial ideals I ⊂ K[x, y] of height 2, generated by 3 elements, the fiber cone F (I) of I is a hypersurface ring, and that F (I) has positive depth for interesting classes of height 2 monomial ideals I ⊂ K[x, y], which are generated by 4 elements. For these classes of ideals we also show that F (I) is Cohen-Macaulay if and only if the defining ideal J of F (I) is generated by at most 3 elements. In all the cases a minimal set of generators of J is determined.
To a simplicial complex ∆ , we associate a square-free monomial ideal F (∆) in the polynomial rin... more To a simplicial complex ∆ , we associate a square-free monomial ideal F (∆) in the polynomial ring generated by its facet over a field. Furthermore, we could consider F(∆) as the Stanley-Reisner ideal of another simplicial complex δN (F(∆)) from facet ideal theory and Stanley-Reisner theory. In this paper, we determine what families of simplicial complexes ∆ have the property that their Stanley-Reisner complexes δN (F(∆)) are shellable. Furthermore, we show that the simplicial complex with the free vertex property is sequentially Cohen-Macaulay. This result gives a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.
The aim of this study was to compare quantitative polymerase chain reaction (qPCR) with immunohis... more The aim of this study was to compare quantitative polymerase chain reaction (qPCR) with immunohistochemistry (IHC) for the detection of Her-2 in gastric cancer, and to investigate the correlation between the expression levels of human epidermal growth factor receptor 2 (Her-2) and clinical features. Clinical data from 426 cases of gastric cancer were collected. Her-2 expression levels in cancerous tissue were detected using IHC, and the Her-2/neu gene expression levels were determined by qPCR. The correlation between the expression level of Her-2 and clinical features was investigated. The positive expression rate of Her-2 in cancerous tissue detected using qPCR and IHC was 11.17% (46/412) and 13.38% (57/426), respectively. The positive expression of the Her-2 protein/gene was significantly correlated with the depth of invasion and lymphatic metastasis, as well as the TNM stage (P<0.05). No significant correlation was identified between positive expression of the Her-2 protein/ge...
Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal with almost mi... more Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal with almost minimal mixed multiplicity such that depth G(I) d − 1. We show that F m (I) has almost maximal depth (i.e. depth F m (I) d − 1).
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