scholarly journals Structure of some numerical semigroup rings

2020 ◽  
Vol 121 ◽  
pp. 53-62
Author(s):  
Philippe Gimenez ◽  
Hema Srinivasan
Keyword(s):  
Numerical Semigroup ◽  
Semigroup Rings ◽  
Numerical Semigroup Rings

ON THE COHEN–MACAULAYNESS OF SOME GRADED RINGS

2008 ◽  
Vol 07 (01) ◽  
pp. 109-128
Author(s):  
D. P. PATIL ◽  
G. TAMONE
Keyword(s):  
Numerical Semigroup ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Associated Graded Ring ◽  
Blowing Up ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Numerical Semigroup Rings

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and [Formula: see text], [Formula: see text]. We assume that the length [Formula: see text]. This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr 𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪s + xR, where s is the degree of the h-polynomial h R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition [Formula: see text].

scholarly journals Quasi-socle ideals in Gorenstein numerical semigroup rings

2008 ◽  
Vol 320 (1) ◽  
pp. 276-293 ◽  
Author(s):  
Shiro Goto ◽  
Satoru Kimura ◽  
Naoyuki Matsuoka
Keyword(s):  
Numerical Semigroup ◽  
Semigroup Rings ◽  
Numerical Semigroup Rings
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