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### Chaudhari, Jayprakash Ninu

- 2-Absorbing Ideals in the Semiring of Non-Negative Integers

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1 Department of Mathematics, M. J. College, Jalgaon-425 002, IN

#### Authors

**Affiliations**

1 Department of Mathematics, M. J. College, Jalgaon-425 002, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 80, No 3-4 (2013), Pagination: 235-241#### Abstract

All 2-absorbing ideals in the semiring of non-negative integers are investigated.#### Keywords

Semiring, Prime Ideal, 2-Absorbing Ideal, Finitely Generated Ideal.- A Remark on n-absorbing Ideals of Principal Ideal Domains

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1 Department of Mathematics M. J. College, Jalgaon - 425 002, IN

#### Authors

**Affiliations**

1 Department of Mathematics M. J. College, Jalgaon - 425 002, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 81, No 3-4 (2014), Pagination: 227-229#### Abstract

Let*R*be a commutative ring with identity 1 ≠ 0. General-izing the notion of prime ideals in

*R*, Anderson and Badawi introduced the notion of n-absorbing ideals in

*R*. We introduce the notion of n- absorbing domains which is a generalization of an integral domain. We investigate all n-absorbing ideals in a PID

*R*and hence prove that Z

_{m}is an n-absorbing domain if and only if m =

*p*

_{1}^{r1}

*p*

_{2}^{r2}

*p*

_{3}^{r3}…

*p*

_{k}^{rk}where k ≤ n and r1 + r2 + r3 + … + rk ≤ n.

#### Keywords

Principal Ideal Domain, n-Absorbing Ideal, n-Absorbing Domain.#### References

- D. F. Anderson and Ayman Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39(2011), 1646 – 1672.
- Ayman Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. Vol. 75(2007), 417 – 429.
- I. N. Herstein, Topics in algebra, Blaisdell Publishing company, 1964.
- Sh. Payrovi and S. Babaei, On the 2-absorbing ideals in commutative rings, Bull.Malays. Math. Sci. Soc. (To appear).

- On
*B*-Ideals in Semirings

*B*-Ideals in Semirings
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1 Department of Mathematics, ACS College, Dharangaon 425 105,, IN

2 Department of Mathematics, M. J. College, Jalgaon 425002, IN

#### Authors

**Affiliations**

1 Department of Mathematics, ACS College, Dharangaon 425 105,, IN

2 Department of Mathematics, M. J. College, Jalgaon 425002, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 1-8#### Abstract

In this paper, we introduce the notion of*B*-ideal in a commutative semiring

*R*. Then 1) A characterization of

*B*-ideals in the Semiring of non-negative integers is obtained. 2) Relation between

*B*-ideals in a semiring

*R*containing a Q-ideal

*I*of

*R*and

*B*-ideals in the quotient semiring R/I

_{(Q)}is obtained. Further study of

*k*-Noetherian semirings is developed. Also

*B*-ideals in polynomial semirings are studied.

#### Keywords

Semiring, Zerosumfree Semiring, Subtractive Ideal, Partitioning Ideal,*B*-ideal, Strong

*B*-ideal, Quotient Semiring,

*K*-Noetherian Semiring, Polynomial Semiring, Monic Ideal, Coefficient Ideal.

#### References

- Paul J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21. (1969), 412-416.
- Shahabaddin Ebrahimi Atani, The ideal theory in quotients of commutative semirings, Glasnik Matematicki, 42. (62)(2007), 301-308.
- J. N. Chaudhari and D. R. Bonde, Ideal theory in quotient semirings, Thai J. Math., 12. (1) (2014), 95-101.
- J. N. Chaudhari, 2-absorbing ideals in semirings, Inter. J. Alg., 6(6)(2012), 265-270.
- J. N. Chaudhari and V. Gupta, Weak primary decomposition theorem for right Noetherian semirings, Indian J. Pure and Appl. Math, 25. (6)(1994), 647-654.
- L. Dale, Monic and monic free ideals in polynomial semiring, Proc. Am. Math. Soc., 56(1) (1976), 45-50.
- A. Y. Darani, On 2-Absorbing and Weakly 2-Absorbing Ideals of Commutative Semirings, Kyungpook Math. J., 52(2012), 91-97.
- J. S. Golan, Semirings and Their Applications, Kluwer, Dordrecht, 1999.
- Vishnu Gupta and J. N. Chaudhari, On partitioning ideals of semirings, Kyungpook Math. J., 46. (2006), 181-184.
- V. Gupta and J. N. Chaudhari, On prime ideals in semirings, Bull. Malaysian Math. Sc. Soc., 34(2) (2011), 417–421.
- T. K. Mukherjee, M. K. Sen and Shamik Ghosh, Chain Conditions on Semirings, Internat. J. Math. and Math. Sci., 19(2)(1996), 321-326.