Strongly Nil* Clean Ideals

An element is known a strongly nil* clean element if , where , are idempotents and is nilpotent, that commute with one another. An ideal of a ring is called a strongly nil* clean ideal if each element of is strongly nil* clean element. We investigate some of its fundamental features, as well as its relationship to the nil clean ideal


INTRODUCTION
In this paper, a ring is associative with unity unless otherwise expressed. are respectively Jacobson radical, the set of unit, idempotent and nilpotent elements of respectively. "An ideal of a unital ring is clean in case every element in is a sum of an idempotent and a unit of " [4]. In [1] Sharma and Basnet defined the concept nil clean ideal (henceforth: NCI) as for each there is a nilpotent element in and an idempotent element then . We call is strongly nil clean ideal (henceforth: SNCI) if each in are written as where and [1]. An element in a ring is called tripotent if [6], "An ideal is called strongly tri nil clean ideal (henceforth: STNCI) if for each element can be expressed as where is tripotent and is nilpotent elements with " [5]. This paper introduces the concept of strongly nil* clean ideal (henceforth: SN*CI).We give some of it's properties, and find it's relationships with NC ideals.

2.
Strongly nil* clean ideals: In this section we introduce the concept of the SN*CI. Some of it's characteristics are discussed as well as some examples.

Definition 2.1:
An element of a ring is said to be strongly nil clean if , where and and . A ring is said to be strongly nil clean if every element of is strongly nil clean [2].

Example 2.2:
The ring of integers modulo 4, Z 4 is SNC ring.

Definition 2.3:
An element of a ring is known SN*C element if for each there exist two idempotent elements 1 , 2 in and a nilpotent element in that commute with one another, such that 1 1 2 . A ring is said to be SN*C ring if each element of is SN*C element.

Example 2.4:
The ring of integers modulo 8, Z 8 is SN*C ring.

Definition 2.5:
An ideal of a ring is known SN*CI if each element of is SN*C element.

Example 2.6:
Consider the ring of integers modulo 16, the ring of Z 16 contained three proper ideals namely: The ideals , and of a ring Z 16 are SN*C ideals.

Proof:
Since ) ) then . Note that: . Hence is an idempotent.

Proposition 2.8:
Let is a SN*CI. Then is a SNCI.

Proof:
Let be strongly nil* clean ideal. Then for all , we have where are idempotent elements and is a nilpotent that commute with one another. By (lemma 2.7) is idempotent. Then we get is idempotent, since ( . Hence and . Hence is strongly nilclean ideal.
Next, we give the following results:

Lemma 2.9:
If , then is a unit.

Рroof:
Since then , for some positive integer . If we set , showing that is a unit. Since , hence . So is also unit.

Proposition 2.10:
If is SN*CI, and if , then is a nilpotent.

Рroof:
Let be SN*CI such that then that commute with one another.

Proposition 2.11:
If is SN*C ring, then is a nil ideal.

Proof:
Suppose then is a unit. Since is SN*C ring, then . Now , this implies . Hence . This implies where is a unit. Thus , but is an idempotent. Then . So . Therefore

Proposition 2.12:
Let be an ideal of a ring with every , if is nilpotent, then is SNCI.

Proof:
Since then is an idempotent we get . Now . Then . Now Thus . On the other hand, . Since . Then is SNCI.

Proposition 2.13:
If is local ring, and is SN*CI of , then is a nil ideal.

Proof:
Let be a local ring, then either or is a unit. Let be a SN*CI of , and let , if is a unit. Then . Let is a unit. Since is a SN*CI, then , where and that commute with one another. Now then .
Since is a unit we get also is unit, say . Then .
Since is idempotent By (lemma2.7). Then is also idempotent. Hence , this implies . Therefore . Hence is a nil ideal.

Lemma 2.14:
Let be a ring, with , and if is idempotent element, then is a unit.

Proof:
Let . Then . Therefore is a unit.

Theorem 2.15:
Let be a ring, with , and be a SN*CI, then each element of , can be written as a sum of two units.

Proof:
Let be a SN*CI and then , Where and , that commute with one another. Consider . Since is an idempotent. Then by (lemma2.14) is a unit, say and is a unit, say , then .

Tri nil clean ideal
In this section we give the definition of the tri nil clean ideal. We investigate some of its properties and provide some examples.

Definition 3.1:
An ideal is known TNCI if for each element can be expressed as where and if further then is called STNCI [5]. Clearly every NCI is TNCI.
The next results shows the relation between TNCI with strongly clean ideal and nil ideals.

Proposition 3.3:
If is an ideal with every . Then is a strongly clean ideal.

Proof:
Let Consider , then . Hence is a unit. This implies Since is a unit. Then where , by (lemma2.9) . We get *, where * , since is an idempotent. Then * * where * . Hence is strongly clean ideal.

Proposition 3.4:
Let be an ideal of a ring and If every element of , where . Then is a nil ideal of .

Proof:
Let where and . Now . Since Then Let . Then Thus is a nil ideal of .

Let
, and let then .

Proof:
Let then then is unit. Let . Since Then it follows , then . So . Hence .

Proposition 3.6:
If is strongly tripotent ideal , if Then is a nil ideal.