• Kyungpook Mathematical Journal
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• 제57권3호
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• pp.401-417
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• 2017

Ever since their introduction, skew PBW ($Poincar{\acute{e}}$-Birkhoff-Witt) extensions of rings have kept growing in importance, as researchers characterized their properties (such as primeness, Krull and Goldie dimension, homological properties, etc.) in terms of intrinsic properties of the base ring, and studied their relations with other fields of mathematics, as for example quantum mechanics theory. Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW extensions. Our aim in this paper is to study skew PBW extensions of Baer, quasi-Baer, principally projective and principally quasi-Baer rings, in the case when the base ring R is not assumed to be reduced. We just impose some mild compatibleness over the base ring R, and prove that these properties are stable over this kind of extensions.

• 대한수학회보
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• 제54권6호
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• pp.1913-1925
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• 2017

Let R be a ring with identity. An ideal N of R is called ideal-symmetric (resp., ideal-reversible) if $ABC{\subseteq}N$ implies $ACB{\subseteq}N$ (resp., $AB{\subseteq}N$ implies $BA{\subseteq}N$) for any ideals A, B, C in R. A ring R is called ideal-symmetric if zero ideal of R is ideal-symmetric. Let S(R) (called the ideal-symmetric radical of R) be the intersection of all ideal-symmetric ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring R, we have $S(M_n(R))=M_n(S(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a quasi-Baer ring R, R is semiprime if and only if R is ideal-symmetric if and only if R is ideal-reversible.

• 대한수학회지
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• 제44권6호
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• pp.1267-1279
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• 2007

For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.