On 2-Absorbing and Weakly 2-Absorbing Primary Ideals of a Commutative Semiring
0
조회
0
다운로드
3
인용
0
참고문헌
키워드
초록
Let $R$ be a commutative semiring. The purpose of this note is to investigate the concept of$2$-absorbing (resp., weakly $2$-absorbing) primary idealsgeneralizing of $2$-absorbing (resp., weakly $2$-absorbing) idealsof semirings. A proper ideal $I$ of $R$ said to be a $2$-absorbing(resp., weakly $2$-absorbing) primary ideal if whenever $a,b,c\in R$such that $abc\in I$ (resp., $0\neq abc\in I$), then either $ab\inI$ or $bc\in \sqrt{I}$ or $ac\in \sqrt{I}$. Moreover, when $I$ is a$Q$-ideal and $P$ is a $k$-ideal of $R/I$ with $I\subseteq P$, it isshown that if $P$ is a $2$-absorbing (resp., weakly $2$-absorbing)primary ideal of $R$, then $P/I$ is a $2$-absorbing (resp., weakly$2$-absorbing) primary ideal of $R/I$ and it is also proved that if$I$ and $P/I$ are weakly $2$-absorbing primary ideals, then $P$ is aweakly $2$-absorbing primary ideal of $R$.
Abstract
Let $R$ be a commutative semiring. The purpose of this note is to investigate the concept of$2$-absorbing (resp., weakly $2$-absorbing) primary idealsgeneralizing of $2$-absorbing (resp., weakly $2$-absorbing) idealsof semirings. A proper ideal $I$ of $R$ said to be a $2$-absorbing(resp., weakly $2$-absorbing) primary ideal if whenever $a,b,c\in R$such that $abc\in I$ (resp., $0\neq abc\in I$), then either $ab\inI$ or $bc\in \sqrt{I}$ or $ac\in \sqrt{I}$. Moreover, when $I$ is a$Q$-ideal and $P$ is a $k$-ideal of $R/I$ with $I\subseteq P$, it isshown that if $P$ is a $2$-absorbing (resp., weakly $2$-absorbing)primary ideal of $R$, then $P/I$ is a $2$-absorbing (resp., weakly$2$-absorbing) primary ideal of $R/I$ and it is also proved that if$I$ and $P/I$ are weakly $2$-absorbing primary ideals, then $P$ is aweakly $2$-absorbing primary ideal of $R$.
참고문헌 (18)
- P. J. Allen, "A fundamental theorem of homomorphisms for semirings", Proc. Amer . Math . Soc., 21, pp.412~416 (1969)
- Paul J. Allen, "Ideal Theory in Commutative A-semirings", Kyungpook Mathematical Journal, 46(2), pp.261~271 (2006), 경북대학교 자연과학대학 수학과
- S. E. Atani, "The ideal theory in quotient of commutative semirings", Glasnik Matem-aticki, 42(62), pp.301~308 (2007)
- S. E. Atani, "On weakly primary ideals", Georgian Math. J., 12(3), pp.423~429 (2005)
- S. E. Atani, "Ideals in quotient semirings", Chiang Mai J. Sci., 40(1), pp.77~82 (2013)
- A. Badawi, "On 2-absorbing ideals of commutative rings", Bull. Austral. Math. Soc., 75, pp.417~429 (2007)
- A. Badawi, "On weakly 2-absorbing ideals of commutative rings", Houston J. Math., 39, pp.441~452 (2013)
- Ayman Badawi, "On 2-absorbing primary ideals in commutative rings", 대한수학회보, 51(4), pp.1163~1173 (2014), 대한수학회
- J. N. Chaudhari, "2-absorbing subtractive ideals in semimodules", Jornal of Advance Research in Pure Math, 5, pp.118~124 (2013)
- J. N. Chaudhari, "A note on strongly Euclidean semirings", Interna-tional Journal of Algebra, 6(6), pp.271~275 (2012)
- J. N. Chaudhari, "On n-absorbing ideals of the semiring Z+0", Journal of Advanced Research in Pure Math, 6, pp.25~31 (2014)
- J. S. Golan, "Semiring and their applications" (1999), Kluwer Academic publisher
- V. Gupta, "Some remark on semirings", Rad. Mat., 12, pp.13~18 (2003)
- Ch. B. Kim, "On the quotient structure of k-semirings", J. Sinc. Institute Kookmin University of Korea, 2, pp.11~16 (1985)
- Ch. B. Kim, "A note on the localization in semirings", J. Sinc. Institute Kookmin University of Korea, 3, pp.13~19 (1985)
- D. R. LaTorre, "A note on quotient semirings", Proc. Amer. Math. Soc., 24, pp.463~465 (1970)
- H. S. Vandive, "Note on simple type of algebra in which the cancellation law of addition does not hold", Bull. Amer. Math. Soc., 40, pp.914~920 (1934)
- Ahmad Yousefian Darani, "On 2-Absorbing and Weakly 2-Absorbing Ideals of Commutative Semirings", Kyungpook Mathematical Journal, 52(1), pp.91~97 (2012), 경북대학교 자연과학대학 수학과