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FUSIBLE RINGS 1. Introduction It Is Well-Known That the Sum of Two Zero

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FUSIBLE RINGS 1. Introduction It Is Well-Known That the Sum of Two Zero FUSIBLE RINGS EBRAHIM GHASHGHAEI AND WARREN WM. MCGOVERN∗ Abstract. A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x; σ] and R[[x; σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semi-simple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full Rings of continuous functions are fusible. It is also shown that if 1 = e1 + e2 + ::: + en in a ring R where the ei are orthogonal idempotents and each eiRei is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings. Keywords: Fusible rings, unit fusible rings, zero-divisors, clean rings, C(X). MSC(2010): Primary:16U99; Secondary:16W99, 13A99. 1. Introduction It is well-known that the sum of two zero-divisors need not be a zero-divisor. In [18, Theorem 1.12] the authors characterized commutative rings for which the set of zero-divisors is an ideal. The set of zero-divisors (i.e., Z(R)) is an ideal in a commutative ring R if and only if the classical ring of quotients of R (i.e., qcl(R)) is a local ring if and only if Z(R) is a prime ideal of R and qcl(R) is the localization at Z(R). We remind the reader that a finite commutative ring R is local if and only if Z(R) is an ideal. It is clear that if the set of left zero-divisors in a ring R is not a left ideal, then there exists a left zero-divisor which can be expressed as the sum of a left zero-divisor and a non-left zero-divisor in R. This leads to our investigation of the class of rings in which every element can be written as the sum of a left zero-divisor and a non-left zero-divisor. Throughout this paper, R will be an associative ring with identity, U(R) its group of units, J(R) its Jacobson radical, Id(R) its set of idempotents, and r(R) the set of regular elements of R. For x 2 R, Annl(x) = fa 2 R : ax = 0g and Annr(x) denote the left annihilator and the right annihilator ideals of x in R, respectively. When Annr(x) 6= f0g we say x is a left zero-divisor; otherwise it is a non-left zero-divisor. ∗ Let Zl(R) (respectively, Zl (R)) denote the set of left zero-divisors (respectively, ∗ non-left zero-divisors) of R. Similarly, let Zr(R) (respectively, Zr (R)) denote the set of right zero-divisors (respectively, non-right zero-divisors) of R. Note that ∗ ∗ Zl (R) \ Zr (R) = r(R). Clearly, for a commutative ring R, Zl(R) = Zr(R) = Z(R) ∗ ∗ and Zl (R) = Zr (R) = r(R). We denote the ring of n × n matrices over R by l r Mn(R). For a ring R, qcl(R) (respectively, qcl(R)) denote the classical (or total) left (respectively, right) ring of quotients of R, when it exists. If R is a commutative l r ring, then qcl(R) = qcl(R) = qcl(R). ∗Corresponding author. 2 GHASHGHAEI AND MCGOVERN Our main goal is to study the class of fusible rings. In the next section, we investigate some fundamental properties of this class. In the third section, we consider the commutative case. In the last section, we strengthen the definition of a fusible element and report our results on this subclass of rings. We end this section by recalling some ring theoretic concepts that will play a pivotal role throughout the article. Notation 1.1. Let R be a ring with identity. (1) The Jacobson radical of R is denoted by J(R). When J(R) = 0, R is said to be Jacobson semi-simple or semi-primitive. (2) R is said to be reduced if it has no nonzero nilpotent elements. (3) An element a 2 R is a left zero-divisor if there is 0 6= r 2 R with ar = 0. An element which is not a left zero-divisor is called a non-left zero-divisor. (Right zero-divisors and non-right zero-divisors are defined analogously.) Notice that 0 2 Zl(R) \ Zr(R). (4) An element a 2 R is regular if it is neither a left zero-divisor nor a right zero-divisor. The set of regular elements of R is denoted by r(R). 2. fusible rings We begin with a formal definition of the central concept of the article. Definition 2.1. We call a nonzero element a 2 R left fusible if it can be expressed as the sum of a left zero-divisor and a non-left zero-divisor in R. We call a ring R left fusible if every nonzero element of R is left fusible. Right fusible rings are defined analogously. A ring which is both right and left fusible is called fusible. A ring R is said to be reversible if ab = 0 implies ba = 0 for a; b 2 R. Exam- ples of reversible rings are symmetric rings, reduced rings, and commutative rings. Therefore in the class of reversible rings the two concepts of left fusibility and right fusibility coincide. Remark 2.2. Observe that a regular element is trivially left fusible; similarly for right fusible. Therefore, every domain is fusible, e.g. Z. Also, every nonzero idempotent is right and left fusible: for each 0 6= e 2 Id(R), e = (1 − e) + (2e − 1) is a fusible representation of e. Therefore, every Boolean ring is fusible. We recall that a ring R is said to be almost clean if each of its elements can be written as the sum of an idempotent and a regular element. Almost clean rings, in the commutative context, were introduced by McGovern in [23]. It is clear that an almost clean representation of an element a = e + r where 1 6= e 2 Id(R) and r 2 r(R), serves as a fusible representation. That 1 6= e is important since not every almost clean ring is fusible. A ring R is said to be special almost clean if each element a can be decomposed as the sum of a regular element r and an idempotent e with aR T eR = 0, see [2]. Thus, every special almost clean ring is fusible. Note that there exist fusible rings which are not special almost clean rings, see [2, Example 4.2] and Remark 5.7. We should remind the reader that there exist almost clean rings which are not special almost clean rings, for example Z4. Example 2.3. For any prime integer p and n ≥ 2, Zpn is not fusible. In these rings the powers of p, pk (1 ≤ k ≤ n − 1) cannot be expressed as the sum of a zero-divisor and a non-zero-divisor in Zpn . It follows that the class of fusible rings FUSIBLE RINGS 3 is not closed under homomorphic images. (Note that it will follow from later work that Zn is fusible if and only if n is square free.) There are some nice properties of rings that force fusibility. We consider a few of these and then move to consider the passage of the condition to polynomial rings. Proposition 2.4. Let R be a ring with comparability relation between right anni- hilators of its elements. Then R is left fusible if and only if R is a domain. Proof. As has already been observed a domain is left fusible. Suppose that R is ∗ left fusible and 0 6= a 2 Zl(R). There exist z 2 Zl(R) and s 2 Zl (R) such that a = z + s. Now by assumption Annr(a) ⊆ Annr(z) or Annr(z) ⊆ Annr(a). In either case any non-zero element belonging to both will also belong to Annr(s). Hence, s is a left zero-divisor, a contradiction. Example 2.5. A ring R is said to be right (left) uniserial if and only if its right (left) ideals are totally ordered under inclusion. Commutative uniserial rings are known as chained rings. It follows that a right uniserial ring is left fusible if and only if it is a domain. Remark 2.6. Satyanarayana in [30, Proposition 2.4] proved that if R is a ring r r with a classical right quotient ring qcl(R), then qcl is a local ring if and only if Z(R) is an ideal. Hence a ring R with a local classical right quotient ring has no fusible right zero-divisor. ∗ Q Q ∗ Lemma 2.7. Let fRigi2I be a family of rings. Then Zl ( i2I Ri) = i2I Zl (Ri) ∗ Q Q ∗ Q Q and Zr ( i2I Ri) = i2I Zr (Ri). Also, U( i2I Ri) = i2I U(Ri). Q Proposition 2.8. Let fRigi2I be a family of rings and let R = i2I Ri. R is fusible if and only for each i 2 I, Ri is fusible. Proof. Suppose that k be an arbitrary element in I and 0 6= ak 2 Rk. Put (xi)i2I 2 Q Q i2I Ri such that xi = ak for i = k, in otherwise xi = 0.
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