## When is the $$2\times 2$$ matrix ring over a commutative local ring strongly clean?(English)Zbl 1110.16029

Throughout all rings are associative with identity. A ring $$R$$ is said to be strongly clean if every element of $$R$$ is the sum of an idempotent and a unit that commute. Local rings are obviously strongly clean. Let $$U(R)$$ be the group of units of a ring $$R$$, $$J(R)$$ be the Jacobson radical of $$R$$ and $$C_2$$ be the cyclic group of order 2.
The main results established by the authors are the following. (1) Let $$R$$ be a commutative ring. If $$M_2(R)$$ is strongly clean, then for all $$w\in J(R)$$ the equation $$x^2-x=w$$ is solvable in $$R$$. (2) Let $$R$$ be a commutative local ring and let $$n\geq 1$$. Then: (i) if either $$2\in U(R)$$ or $$R/J(R)\cong\mathbb{Z}_2$$, then $$M_2(R)$$ is strongly clean if and only if for all $$w\in J(R)$$ the equation $$x^2-x=w$$ is solvable in $$R$$; (ii) if $$2\in J(R)$$ and $$R/J(R)\not\cong\mathbb{Z}_2$$, then $$M_2(R)$$ is strongly clean if and only if for all $$w_1,w_2\in J(R)$$ the equation $$x^2+(1+w_1)x=w_2$$ is solvable in $$R$$; (iii) $$M_2(R)$$ is strongly clean if and only if $$M_2(R[x]/(x^n))$$ is strongly clean if and only if $$M_2(RC_2)$$ is strongly clean.

### MSC:

 16S50 Endomorphism rings; matrix rings 16U60 Units, groups of units (associative rings and algebras)
Full Text:

### References:

 [1] Burgess, W.D.; Menal, P., On strongly π-regular rings and homomorphisms into them, Comm. algebra, 16, 1701-1725, (1988) · Zbl 0655.16006 [2] Camillo, V.P.; Khurana, D., A characterization of unit regular rings, Comm. algebra, 29, 5, 2293-2295, (2001) · Zbl 0992.16011 [3] Camillo, V.P.; Yu, H.P., Exchange rings, units and idempotents, Comm. algebra, 22, 4737-4749, (1994) · Zbl 0811.16002 [4] J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005 [5] Dischinger, M.F., Sur LES anneaux fortement π-réguliers, C. R. acad. sci. Paris Sér. A, 283, 571-573, (1976) · Zbl 0338.16001 [6] Han, J.; Nicholson, W.K., Extensions of Clean rings, Comm. algebra, 29, 2589-2595, (2001) · Zbl 0989.16015 [7] Khurana, D.; Lam, T.Y., Clean matrices and unit-regular matrices, J. algebra, 280, 2, 683-698, (2004) · Zbl 1067.16050 [8] Lam, T.Y., A first course in noncommutative rings, (2001), Springer New York · Zbl 0980.16001 [9] Nicholson, W.K., Local group rings, Canad. math. bull., 15, 137-138, (1972) · Zbl 0235.16008 [10] Nicholson, W.K., Lifting idempotents and exchange rings, Trans. amer. math. soc., 229, 269-278, (1977) · Zbl 0352.16006 [11] Nicholson, W.K., Strongly clean rings and Fitting’s lemma, Comm. algebra, 27, 3583-3592, (1999) · Zbl 0946.16007 [12] Nicholson, W.K.; Varadarajan, K.; Zhou, Y., Clean endomorphism rings, Arch. math. (basel), 83, 4, 340-343, (2004) · Zbl 1067.16051 [13] O’Meara, K.C., The exchange property for row and column-finite matrix rings, J. algebra, 268, 744-749, (2003) · Zbl 1046.16012 [14] E. Sánchez Campos, On strongly clean rings, unpublished [15] Wang, Z.; Chen, J., On two open problems about strongly Clean rings, Bull. austral. math. soc., 70, 279-282, (2004) · Zbl 1069.16035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.