### Article

Kyungpook Mathematical Journal 2022; 62(3): 425-436

**Published online** September 30, 2022

Copyright © Kyungpook Mathematical Journal.

### On 2-absorbing Primary Ideals of Commutative Semigroups

Manasi Mandal∗ and Biswaranjan Khanra

Department of Mathematics, Jadavpur University, Kolkata-700032, India

e-mail : manasi_ju@yahoo.in and biswaranjanmath91@gmail.com

**Received**: January 5, 2021; **Revised**: April 24, 2022; **Accepted**: May 3, 2022

### Abstract

In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

**Keywords**: Commutative semigroup, Prime ideal, Maximal ideal, Semiprime ideal, Semiprimary ideal, 2-absorbing ideal

### 1. Introduction

The concept of a 2-absorbing ideal for a commutative ring was introduced by Badawi [2] and later extended to commutative semigroups by Cay et al. [4] as follows : a proper ideal

In this paper, we define a 2-absorbing primary ideal in a commutative semigroup (

Before going to the main work we discuss some preliminaries which are necessary:

**Definition 1.1.** ([13]) A non-empty ideal

**Remark 1.2.** These concepts coincide if

**Definition 1.3.** ([11]) For an ideal

**Definition 1.4.** ([9]) An ideal

**Definition 1.5.** ([11]) A commutative semigroup

**Definition 1.6.** ([9]) A commutative semigroup

**Definition 1.7.** ([13]) An ideal _{1}

**Theorem 1.8.** ([11]) If

(1)

(2)

(3)

(4)

(5) If

**Definition 1.9.**([13]) An ideal

**Definition 1.10.** ([5]) Let

### 2. Some Properties of 2-absorbing Primary Ideals

Throughout this paper, unless otherwise mentioned,

**Definition 2.1.** A proper ideal

Since

**Theorem 2.2.** Let

The following lemmas are obvious, hence we omit the proof.

**Lemma 2.3.**([Lemma

**Lemma 2.4.**([Theorem

**Lemma 2.5.**([Lemma _{1}_{2}_{1} ∩ P_{2}

**Corollary 2.6.** Let

(1) if _{1}_{2}

(2) every maximal ideal of

(3) every prime ideal of

**Remark 2.7.** The following example shows that converse of Lemma

**Remark 2.8.** The following example shows that converse of Theorem 2.2 is not true. Consider the ideal

A semigroup

**Corollary 2.9.** Let

The following is a characterization of a semigroup in which 2-absorbing primary ideals are prime:

**Theorem 2.10.** Let

_{1}_{2}

Conversely, Let

Since every primary ideals of a commutative semigroup

**Corollary 2.11.** Let

(

(

The following is a characterization of a semigroup in which 2-absorbing primary ideals are maximal:

**Theorem 2.12.** Let

Conversely, if

Moreover, we prove that

**Corollary 2.13.** Let

**Theorem 2.14.** Let

Case(1). Suppose

Case(2). Suppose

Therefore

The following are obvious consequence of above theorem:

**Corollary 2.15.** Let

(1) if

(2) if

(3) every primary ideal of

**Remark 2.16.** The converse of Theorem 2.14 is not true. Consider the principal ideal

The following theorem is a characterization of a semigroup in which 2-absorbing primary ideals are semiprimary:

**Theorem 2.17.** Let

(1) 2-absorbing primary ideals of

(2) Prime ideals of

(3)

(4) Semiprime ideals are linearly ordered.

(5) Semiprime ideals of

_{1}_{2}

_{1}_{2}

_{1}_{2}

**Definition 2.18.** A commutative semigroup

**Example 2.19.** Consider the commuttive semigroup

**Theorem 2.20.** Let

_{1}_{2}

**Theorem 2.21.** Let

**Theorem 2.22.** Let

(1) 2-absorbing primary ideals of

(2) prime ideals of

(3) idempotents of

(4) All ideals of

(5)

(6)

(7)

(8) 2-absorbing ideals of

(7)

**Lemma 2.23.** Let

**Theorem 2.24.** Let

(1)

(2) If

(3) If

(4)

(5)

(2) Let

(5) Let

**Theorem 2.25.** Let

Conversely, let

Clearly arbitary union of 2-absorbing primary ideals of a semigroup

**Example 2.26.** Let _{1}_{2}

Let

**Theorem 2.27.** Each ρ-class of

_{1}_{2}_{n}_{i}

Therefore the semigroup

**Theoprerm 2.28.** Let _{1}_{2}_{1}_{2}_{1}_{2}

**Proposition 2.29.** Let

(1). For every ideals

(2). For every ideals

Let

**Lemma 2.30.** Let

Again let

**Proposition 2.31.** Let

Conversely, let

The following result is a simple consequence of Proposition 2.31:

**Corollary 2.32.** Let

Let

The composition on

**Theorem 2.33.** Let

**Lemma 2.34.** Let

(1)

(2) If

**Theorem 2.35.** Let

(3) If

(2) The proof of

(3) It is trivial.

As a simple consequence of above theorem, we have the following result

**Corollary 2.36.**

**Lemma 2.37.** Let _{i}

(1) If _{1}_{1}

(2) If _{2}_{2}

**Theorem 2.38.** Let _{i}

(1) _{1}_{1}

(2) _{2}_{2}

_{1}_{1}

Conversely, Suppose _{1}_{1}_{1}_{1}

(2) The proof is similar to (1).

**Theorem 2.39.** Let _{i}_{1}_{2}_{1}_{2}_{1}_{2}_{1}_{2}

_{2}_{1}_{1}_{2}_{2}

**Remark 2.40.** The following example shows that converse of Theorem 2.39 is not true. Consider the 2-absorbing primary ideals

**Theorem 2.41.** Let _{1}_{2}

(1)

(2) Either _{1}_{1}_{2}_{2}_{1}_{1}_{2}_{2}

### Acknowledgements.

The authors express their deep gratitude to learned referees for their meticulous reading and valuable suggestions which have definitely improved the paper. The research work reported here is supported by UGC-JRF NET Fellowship (Award No. 11-04-2016-421922) to Biswaranjan Khanra by the University Grants Commission, Governtment of India.

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