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

• Kyungpook Mathematical Journal
• /
• 제58권2호
• /
• pp.203-219
• /
• 2018

This paper is devoted to the study of strongly ${\alpha}-semicommutative$ rings, a generalization of strongly semicommutative and ${\alpha}-rigid$ rings. Although the n-by-n upper triangular matrix ring over any ring with identity is not strongly ${\bar{\alpha}}-semicommutative$ for $n{\geq}2$, we show that a special subring of the upper triangular matrix ring over a reduced ring is strongly ${\bar{\alpha}}-semicommutative$ under some additional conditions. Moreover, it is shown that if R is strongly ${\alpha}-semicommutative$ with ${\alpha}(1)=1$ and S is a domain, then the Dorroh extension D of R by S is strongly ${\bar{\alpha}}-semicommutative$.

• 대한수학회논문집
• /
• 제26권3호
• /
• pp.339-348
• /
• 2011

A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c ${\in}$ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/$\sqrt{I}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then $T_n$(R) and R[x]=($x^n$) are radically-symmetric, where ($x^n$) is the ideal of R[x] generated by $x^n$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (${\alpha}$, ${\delta}$)-compatible ring, then R[x; ${\alpha}$, ${\delta}$] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].

• 대한수학회논문집
• /
• 제39권2호
• /
• pp.373-397
• /
• 2024

Lambek introduced the concept of symmetric rings to expand the commutative ideal theory to noncommutative rings. In this study, we propose an extension of symmetric rings called strongly α-symmetric rings, which serves as both a generalization of strongly symmetric rings and an extension of symmetric rings. We define a ring R as strongly α-symmetric if the skew polynomial ring R[x; α] is symmetric. Consequently, we provide proofs for previously established outcomes regarding symmetric and strongly symmetric rings, directly derived from the results we have obtained. Furthermore, we explore various properties and extensions of strongly α-symmetric rings.

• Bulletin of the Korean Chemical Society
• /
• 제8권3호
• /
• pp.144-145
• /
• 1987

The relative stability as function of geometry in the rigid tricyclopropylcarbinyl cations with varied bond angle (${\alpha}$) between the plane of cyclopropane ring and the bond connecting cyclopropane ring to cationic carbon was examined by $^{19}F$ nmr spectroscopy. 7-p-Fluorophenyltricyclo[2.2.2.$0^{2,6}$]octan-7-yl(4) and 8-p-fluorophenyltricyclo[3.2.2.$0^{2,7}$]nonan-8-yl cation (8) were generated from corresponding tertiary alcohols under stable ion conditions, and their $^{19}F$ chemical shifts were compared with those of model compounds such as 7-nortricyclyl cation (3) and tricyclo[3.3.1.$0^{2,7}$]octan-8-yl cation (7). Consequently, it is concluded that the varied orientation of bond angle (${\alpha}$) within in the bisected conformation does not affect degree of the charge delocalization into cyclopropane ring.

• Mass Spectrometry Letters
• /
• 제4권4호
• /
• pp.79-82
• /
• 2013

Various types of alkaloids observed in several herbal medicines were analyzed by electrospray ionization tandem mass spectrometry in positive ion mode. In the present study, MS/MS spectralpatterns were investigated for eight-types of alkaloids (aporpine, protoberberine, tetrahydroprotoberberine, benzylisoquinoline, protopine, phthalide, morpine, and bisbenzylisoquinoline). For aporpine- and protoberberine-type alkaloids, main fragmentations occurred at substituted groups on rigid ring structures, not showing ring fusion. Interesting fragmentations due to iminolization and retro-Diels-Alder (RDA) reaction were observed in MS/MS spectra of protopine- and tetrahydroprotobereberine-type alkaloids. Also, several types of fragmentations such as inductive cleavage and ${\alpha}$-cleavage, or bond cleavage between two ring structures were observed depending on their structural characteristics. These fragmentation patterns are expected to allow instant classification of the specific alkaloid type in various MS/MS spectra of alkaloids.

• BMB Reports
• /
• 제36권6호
• /
• pp.552-557
• /
• 2003

Oligomers with $\alpha$-aminooxy acids are reported to form very stable turn and helix structures, and they are supposed to be useful peptidomimetics for drug design. A recent report suggested that homochiral oxa-peptides form a strong eight-member-ring structure by a hydrogen bond between adjacent aminooxy-acid residues in a $CDCl_3$ solution. In order to design an $\alpha$-MSH analog with a stable turn conformation, we synthesized four tetramers and one pentamer, based on $\alpha$-MSH sequence, and determined the solution structures of the molecules by two-dimensional NMR spectroscopy and simulated annealing calculations. The solution conformations of the three peptidomimetic molecules (TLV, TDV, and TLL) in DMSO-$d_6$ contain a stable 7-membered-ring structure that is similar to a $\gamma$-turn in normal peptides. Newly-designed tetramer TDF and pentamer PDF have a ball-type rigid structure that is induced by strong hydrogen bonds between adjacent amide protons and carbonyl oxygens. In conclusion, the aminooxy acids, easily prepared from natural or unnatural amino acids, can be employed to prepare peptidomimetic analogues with well-defined turn structures for pharmaceutical interest.