Unit Regular Clean Rings

A ring R is called unit regular clean, if every element is the sum of an idempotent and a unit regular elements. In this paper we introduce the notion of unit regular clean ring. we investigate some of it’s basic properties and it’s relation with clean ring.

Clearly unit regular rings and clean rings are -clean. we also provide an example of ur-clean ring which is not clean. In this work we give some properties of ur-clean rings and its relation with clean ring.

2-Unit regular clean ring
In this section we introduce the notion of unit regular clean ring, we give some of it's properties and provide some examples. Definition 2.1 An element of a ring is unit regular clean, (briefly, -clean) if = + where ∈ ( ) and ∈ ( ). A ring is -clean if each of its elements is -clean.
Clearly, unit regular rings and clean rings are -clean. but the converse is not necessarily be true. as the following example shows.

Example 2.2
The ring of integers, Modulo 4, 4 is not unit regular because 2 is not unit regular in 4. However it is easy to check that 4 is -clean. In general -clean is not necessarily be clean see [11,Theorem 4.1].
Next ,we shall give part of basic properties of -clean rings.

Proposition 2.3:
If is a ring, then ∈ is -clean element if and only if (1 − ) is -clean element.
Proof: Let is -clean element then = + where, ∈ ( ) and ∈ ( ), then (1 − e) is an idempotent and −a is unit regular which implies that is ur-clean element.∎ Note that, for any ring R, and any ideal I of R , if R/I is -clean then is not necessarily to beclean as the following examples shows.

Example 2.4:
1-If is prime number then Z/p ≅ Zp is -clean, but the ring is not clean. 2-The ring of integers modulo 12, 12. Let I = {0, 3, 6, 9} be an ideal of 12. Now 12/ is -clean since 12/ is a field; but 12 is not -clean ring. , idempotent can be lifted modulo, as one sided ideal I of a ring .if for ∈ with − 2 ∈ I, there exists an idempotent ∈ such that e − x ∈ I.
The following result, gives a sufficient condition for to be -clean Theorem 2.5: Let I ⊆ J(R) be any ideal, of a ring then is -clean if and only if the quotient ring R/I is -clean and idempotent lift modulo I.
Clearly, (e + I) is an idempotent element of R/I and So (a + I) is unit regular then / is -clean ring.
Conversely: Suppose that the quotient ring R/I is -clean and idempotent lift modulo I and let be any element in . since R/I is ur-clean we can Write r + I = x + e + I for some unit regular + , and idempotent lift modulo , we assume is an idempotent of the ring , since − + = + is unit regular element of R/I. So − is unit regular of , it follows that may be written as the sum of idempotent and unit regular of by writing , r = (r − e) + e , This proves the sufficiency.∎ If is unit regular element then = + and R ∩ e R = 0. Theorem 2.7: Let be abelian -clean ring ,for any ∈ there exists an idempotent , such that is idempotent.
Proof: Since is -clean, then x = e + a where e is idempotent and is unit regular then a = e + u and R ∩ e R = 0 , then e = 0. So ex = ee + , then ex = e. e , since and are central idempotents, then is idempotent. ∎ In [1] Ashrafi proved that "if R be an abelian r-clean ring, then is also r-clean ring". We do like wise of urclean ring.

Theorem 2.8: Let R be an abelian -clean ring then
is also urclean ring.
Proof: Let ∈ ⊆ R, then = + and e r = re where e is idempotent and ∈ ( ) where is -clean.
Since ∈ , then a = ee e + ′ , it follows that = e e + ′ we want to show that is unit regular and is an idempotent. Then is unit regular, implies that is -clean ring. ∎ Theorem 2.9: Let be a ring with every ∈ R there is b ∈ R, such that a + b ∈ J(R) and . = a, Then R is ur-clean Proof: Let ∈ R, then there is b ∈ R such that a + b ∈ J(R) and . = a Therefore is unit regular. If we write = 0 + , then is -clean.∎ Theorem 2.10: Let R be a ring with every in R there is in such that + is unit and . = 0, Then is reduced -clean ring.  If we set = 0 + then is −clean.∎

3-The relation between -clean and clean rings
In this section we give the relationship between -clean and clean rings. Clearly every clean ring is ur-clean ring since unit is unit regular, but the converse is not necessarily be true. Proof: Since every abelian regular ring is clean then for each ∈ , can be written as = + + where , ∈ ( ) and ∈ ( ) Now since , are orthogonal then = + ∈ ( ) and hence = + which shows that is clean.∎

Theorem 3.3:
If is a directly finite -clean ring, and 0 and 1, are the only idempotents in , then is clean.
Proof: Since is -clean ring, each ∈ can be written as = + , where is a unit regular element and is an idempotent element of R.
If ≠ 0, then there exists ∈ such that = . Thus an idempotent element of .
Thus, is a unit of R. So is clean element, and hence is clean ring.∎ Theorem 3.4: Let R be abelian ring and for every ∈ , there exists ∈ such that + ∈ ( ) and . = 0, then there is in R such that is clean element.
Proof: Let ∈ , and + ∈ ( ) then + = . where is idempotent and is unit. Now So + = and hence = 1 − + but is idempotent say so = (1 − ) + This means that is clean element and hence it is -clean.∎