• 대한수학회논문집
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• 제21권4호
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• pp.641-652
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• 2006

In [2], they found some natural generators for the ideals of the finite ring $Z_{pm}$[X]/$(X^n\;-\;1)$, where p and n are relatively prime. If p and n are not relatively prime $X^n\;-\;1$ is not a product of basic irreducible polynomials but a product of primary polynomials. Due to this fact, to consider the ideals of $Z_{pm}$[X]/$(X^n\;-\;1)$ in 'inseparable' case we need to look at the primary ideals of $Z_{pm}$[X]. In this paper, we find a set of generators of ideals of $Z_{pm}$[X]/(f) for some primary polynomials f of $Z_{pm}$[X].

• 호남수학학술지
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• 제27권3호
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• pp.399-404
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• 2005

Let M be a finite commutative nilpotent algebra over a perfect field k of prime characteristic p and let $M^p$ be the sub-algebra of M generated by $x^p$, $x{\in}M$. Eggert[3] conjectures that $dim_kM{\geq}pdim_kM^p$. In this paper, we show that the conjecture holds for $M=R^+/I$, where $R=k[X_1,\;X_2,\;{\cdots},\;X_t]$ is a polynomial ring with indeterminates $X_1,\;X_2,\;{\cdots},\;X_t$ over k and $R^+$ is the maximal ideal of R generated by $X_1,\;X_2,{\cdots},\;X_t$ and I is a monomial ideal of R containing $X_1^{n_1+1},\;X_2^{n_2+1},\;{\cdots},\;X_t^{n_t+1}$ ($n_i{\geq}0$ for all i).

• 대한수학회보
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• 제44권2호
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• pp.233-239
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• 2007

Let $\mathbb{A}=\mathbb{F}_q$[T] be the polynomial ring over a finite field $\mathbb{F}_q$[T] and K=$\mathbb{F}_q$(T) its field of fractions. Let ${\ell}$ be a fixed prime divisor of q-1. Let J be a finite set of monic irreducible polynomials $P{\in}{\mathbb{A}}$ with deg $P{\equiv}0$ (mod ${\ell})$. In this paper we define the group $C_K$ of circular units in K=k$(\{\sqrt[{\ell}]P\;:\;P{\in}J\})$ in the sense of Kucera [4] and compute the index of $C_K$ in the full unit group $O^*_K$.

• 대한수학회논문집
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• 제36권2호
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• pp.209-227
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• 2021

In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring R middle right-(resp. left-)nil symmetric (mr-nil (resp. ml-nil) symmetric, for short) if abc = 0 implies acb = 0 (resp. bac = 0) for a, c ∈ R and b ∈ nil(R) where nil(R) is the set of all nilpotent elements of R. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce left (resp. right) N-reversible ideal I of a ring R if for any a ∈ nil(R), b ∈ R, being ab ∈ I implies ba ∈ I (resp. b ∈ nil(R), a ∈ R, being ab ∈ I implies ba ∈ I). A ring R is called left (resp. right) N-reversible if the zero ideal is left (resp. right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.