Throughout this paper, rings are associative with identity. Let $ R $ be a ring. The set of all units and the Jacobson radical of $ R $ are denoted by $ U(R) $ and $ J(R) $, respectively. The symbol $ M_{n}(R) $ stands for the $ n\times n $ matrix ring over $ R $ whose identity element is written as $ I_{n} $. The commutant and double commutant of $ a\in R $ are denoted respectively by $ {\rm comm}(a) = \{x\in R: ax = xa\} $ and $ {\rm comm}^2(a) = \{x\in R: yx = xy $ for any $ y\in {\rm comm}(a)\} $.

Rings in which every element is the sum of certain special elements were frequently studied in ring theory. A ring $ R $ is called strongly clean if all of its elements are a sum of an idempotent and a unit which commutes (see [1]). Following Diesl [2], a ring $ R $ is strongly nil clean if for every element $ a $ of $ R $, there exists $ e^2 = e\in R $ and a nilpotent $ b\in R $ such that $ a = e+b $ and $ e\in {\rm comm}(a) $. Chen et al. [3] introduced the concept of a perfect J-clean ring. A ring $ R $ is perfectly J-clean if for each $ a\in R $, there exists $ e^2 = e\in {\rm comm}^2(a) $ such that $ a-e\in J(R) $. Recall that $ a\in R $ is quasi-nilpotent if $ 1-ax\in U(R) $ for any $ x\in $ $ {\rm comm} $$ (a) $, and the set of all quasi-nilpotent elements of $ R $ is denoted by $ R^{\rm qnil} $. It is clear that all nilpotent elements and $ J(R) $ are contained in $ R^{\rm qnil} $. As a generalization of strongly nil clean rings and perfectly J-clean rings, Cui and Yin [4] introduced the notion of strongly quasi-nil clean rings. A ring $ R $ is strongly quasi-nil clean if for every $ a\in R $ there exists $ e^2 = e\in {\rm comm}^2(a) $ such that $ a-e\in R^{\rm qnil} $, it was shown that $ M_n(R) $ was not strongly quasi-nil clean for any ring $ R $ and any $ n\geq 2 $.

In this paper, we determine when a single matrix $ A\in M_2(R) $ is strongly quasi-nil clean where $ R $ is a commutative local ring. It is shown that over a commutative local ring $ R $, a matrix $ A\in M_2(R) $ is strongly quasi-nil clean if and only if either $ A\in (M_2(R))^{\rm qnil} $ or $ I_2-A\in (M_2(R))^{\rm qnil} $ or the equation $ x^2-({\rm tr}A)x+{\rm det}A = 0 $ has a root in $ J(R) $ and a root in $ 1+J(R) $.

We begin with some primary facts, which will be used freely.

Lemma 2.1 [4, Proposition2.1] Let $ R $ be a ring and $ u\in U(R) $. Then

$ (1) $ $ a\in R^{\rm qnil} $ is strongly quasi-nil clean.

$ (2) $ $ a\in R $ is strongly quasi-nil clean if and only if so is $ u^{-1}au $.

$ (3) $ $ a\in U(R) $ is strongly quasi-nil clean if and only if $ 1-a\in R^{\rm qnil} $.

According to [5], an element $ a\in R $ is called quasipolar if there exists $ p^2 = p\in R $ such that $ p\in {\rm comm}^2(a) $, $ a+p\in U(R) $, and $ ap\in R^{\rm qnil} $. Any idempotent $ p $ satisfying the above conditions is called a spectral idempotent of $ a $, which is uniquely determined by the element of $ a $, and is denoted by $ a^{\pi} $. By [4, Proposition2.3], every strongly quasi-nil clean element is quasipolar. In addition, if $ a = e+q $ is a strongly quasi-nil expression of $ a\in R $, then $ a $ is quasipolar with $ 1-e $ the spectral idempotent.

Corollary 2.2 Let $ R $ be a ring and $ A\in M_2(R) $. Then $ A $ is quasi-nilpotent if and only if $ A $ is quasipolar and $ A^{\pi} = I_2 $.

Let $ R $ be a commutative ring. For a square matrix $ A\in M_2(R) $, the notation $ {\rm det}A $ and $ {\rm tr}A $ denoted the determinant and the trace of $ A $, respectively.

Lemma 2.3 [6, Lemma4.1] Let $ R $ be a commutative local ring. The following statements are equivalent for $ A\in M_2(R) $.

$ (1) $ $ A\in (M_2(R))^{\rm qnil} $.

$ (2) $ $ {\rm det}A\in J(R) $ and $ {\rm tr}A\in J(R) $.

$ (3) $ $ A^2\in J(M_2(R)) $.

Lemma 2.4 [7, Lemma4.4] Let $ R $ be a commutative local ring and $ E\in M_2(R) $. Then $ E^2 = E $ if and only if $ E $ is one of the cases of the following statements:

$ (1) $ $ E = 0 $ or $ E = I_2 $.

$ (2) $ $ E = \left( \begin{smallmatrix} a & b \\ c & 1-a \\ \end{smallmatrix} \right) $ with $ bc = a-a^2 $.

It is easy to see that $ A $ is a unit if and only if $ {\rm det}A\in U(R) $, $ A $ is quasinilpotent if and only if $ {\rm det}A\in J(R) $ and $ {\rm tr}A\in J(R) $. Naturally we may consider the case of $ {\rm det}A\in J(R) $ and $ {\rm tr}A\in U(R) $.

Proposition 2.5 Let $ R $ be a commutative local ring and $ A\in M_2(R) $ satisfies that $ A\notin (M_2(R))^{\rm qnil} $ and $ {\rm det}A\in J(R) $. The following statements are equivalent.

$ (1) $ $ A $ is strongly quasi-nil clean.

$ (2) $ $ A $ is similar to a matrix $ \left( \begin{smallmatrix} j & 0 \\ 0 & u \\ \end{smallmatrix} \right) $ with $ j\in J(R), u\in 1+J(R) $.

$ (3) $ The characteristic equation of $ A $, $ x^2-{\rm tr}Ax+{\rm det}A = 0 $ has a root in $ J(R) $ and a root in $ 1+J(R) $.

Proof (1)$ \Rightarrow $(2) Assume that $ A $ is strongly quasi-nil clean. It is clear that $ {\rm tr}A\in U(R) $. By Lemma 2.4, we may let $ E\doteq I_2-A^{\pi} = \left( \begin{smallmatrix} a & b \\ c & 1-a \\ \end{smallmatrix} \right)\in M_2(R) $, where $ bc = a-a^2 $. As $ R $ is local, either $ a\in U(R) $ or $ 1-a\in U(R) $. Let $ U_1 = \left( \begin{smallmatrix} b & a \\ -a & c \\ \end{smallmatrix} \right) $ and $ U_2 = \left( \begin{smallmatrix} a-1 & b \\ c & 1-a \\ \end{smallmatrix} \right) $. Since $ {\rm det}U_1 = a, {\rm det}U_2 = a-1 $, we have at least one of $ U_1 $ and $ U_2 $ which is invertible. We may as well suppose that $ U_1 $ is invertible. We write $ B = ( b_{ij}) = U_1^{-1}AU_1 $. Then $ B $ is strongly quasi-nil clean with $ E^{'}\doteq I_2-B^{\pi} = U_1^{-1}EU_1 = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \\ \end{smallmatrix} \right) $. As $ E^{'}\in {\rm comm}(B) $, we have $ b_{12} = b_{21} = 0 $, which implies that $ B $ is diagonal. Since $ {\rm det}B = {\rm det}A\in J(R) $ and $ {\rm tr}B = {\rm tr}U_1^{-1}AU_1 = {\rm tr}A\in U(R) $, so $ b_{11}\in J(R), b_{22}\in U(R) $ or $ b_{11}\in U(R), b_{22}\in J(R) $. We may as well suppose that it is the first case. Let $ b_{11} = j\in J(R) $ and $ b_{22} = u\in U(R) $. From Lemma 2.4, we have $ E^{'} = \left( \begin{smallmatrix} x & y \\ z & 1-x \\ \end{smallmatrix} \right) $, where $ yz = x-x^2 $. Then $ BE^{'} = E^{'}B $ implies that $ y = z = 0 $, so $ x = x^2 $ is an idempotent. Since $ R $ is local, it follows that $ x = 0 $ or $ x = 1 $. If $ x = 1 $, we have $ B-E^{'} = \left( \begin{smallmatrix} j-1 & 0 \\ 0 & u \\ \end{smallmatrix} \right) \in U(M_2(R))\bigcap (M_2(R))^{\rm qnil} = \emptyset $. Hence $ x = 0 $ and $ B-E^{'} = \left( \begin{smallmatrix} j & 0 \\ 0 & u-1 \\ \end{smallmatrix} \right) \in ((M_2(R)))^{\rm qnil} $. By Lemma 2.3, $ (u-1)^2\in J(R) $, thus $ u-1\in J(R) $, and $ B = \left( \begin{smallmatrix} j & 0 \\ 0 & u \\ \end{smallmatrix} \right) $ where $ j\in J(R), u\in 1+J(R) $.

(2) $ \Rightarrow $ (3) is clear.

(3) $ \Rightarrow $ (1) Let $ A = \left( \begin{smallmatrix} a & b \\ c & d\\ \end{smallmatrix} \right) \in M_2(R) $ with $ {\rm tr}A\in U(R) $ and $ {\rm det}A\in J(R) $. Assume that $ \lambda_1 $ and $ \lambda_2 $ are two roots of the equation $ x^2-({\rm tr}A)x+{\rm det}A = 0 $, where $ \lambda_1\in J(R), \lambda_2\in 1+J(R) $. We note that $ {\rm tr}A\in U(R) $. So we may let $ a\in U(R) $. We write $ V = \left( \begin{smallmatrix} b & a-\lambda_1 \\ \lambda_1-a & c\\ \end{smallmatrix} \right) $. Then $ V\in U(M_2(R)) $ because of $ {\rm det}V = bc-(a-\lambda_1)(\lambda_1-a) = a{\rm tr}A+(\lambda_1^2-2a\lambda_1-{\rm det}A)\in U(R) $. Since $ \lambda_1^2-({\rm tr}A)\lambda_1+{\rm det}A = 0 $, we have $ V^{-1}AV = \left( \begin{smallmatrix} \lambda_1 & 0\\ 0& \lambda_2\ \end{smallmatrix} \right)\doteq B $. We assume that $ \lambda_2 = 1+j $ for some $ j\in J(R) $. Writing $ E = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \\ \end{smallmatrix} \right) $, then $ E^2 = E $ and $ B = E+\left( \begin{smallmatrix} \lambda_1 & 0 \\ 0 & j\\ \end{smallmatrix} \right)\in ((M_2(R)))^{\rm qnil} $. For any $ C = (c_{ij})\in M_2(R) $, if $ CB = BC $, then $ c_{12} = c_{21} = 0 $, so $ EC = CE $. Thus $ E\in {\rm comm}^2(B) $. Therefore $ B $ is strongly quasi-nil clean. By Lemma 2.1, $ A $ is strongly quasi-nil clean.

As a consequence of the above results, we obtain the following conclusion about $ 2\times2 $ matrixes over commutative local rings to be strongly quasi-nil clean.

Theorem 2.6 Let $ R $ be a commutative local ring and let $ A\in M_2(R) $. Then $ A $ is strongly quasi-nil clean if and only if one of the following statements holds:

$ (1) $ $ A\in (M_2(R))^{\rm qnil} $ or $ I_2- A\in (M_2(R))^{\rm qnil} $.

$ (2) $ The equation $ x^2-({\rm tr}A)x+{\rm det}A = 0 $ has a root in $ J(R) $ and a root in $ 1+J(R) $.

Combining Theorem 2.6 with Lemma 2.3, we have the following corollary:

Corollary 2.7 Let $ R $ be a commutative local ring. The following are equivalent for $ A\in M_2(R) $:

$ (1) $ $ A\in M_2(R) $ is strongly quasi-nil clean.

$ (2) $ One of the following statements holds:

$ (i) $ $ A^2\in J(M_2(R)) $ or $ (I_2- A)^2\in J(M_2(R)) $.

$ (ii) $ The equation $ x^2-({\rm tr}A)x+{\rm det}A = 0 $ has a root in $ J(R) $ and a root in $ 1+J(R) $

Corollary 2.8 Let $ R $ be a commutative local ring and $ A\in M_2(R) $. If $ A $ is strongly quasi-nil clean, then one of the following statements holds:

$ (1) $ $ A $ is quasipolar and $ A^{\pi} = 0 $ or $ A^{\pi} = I_2 $.

$ (2) $ $ A $ is quasipolar and $ A^{\pi} $ is similar to a matrix $ \left( \begin{smallmatrix} 1 & 0 \\ 0 & 0 \\ \end{smallmatrix} \right) $.

By Theorem 2.6, we can figure out all strongly quasi-nil clean elements of a ring $ M_2( \mathbb{Z}_2) $.

Example 2.9 Let $ R = M_2( \mathbb{Z}_2) $. Then all elements of $ R $ are strongly quasi-nil except for $ \left( \begin{smallmatrix} 0 & 1\\ 1 & 0 \\ \end{smallmatrix} \right) $ and $ \left( \begin{smallmatrix} 0 & 1 \\ 1 & 1 \\ \end{smallmatrix} \right) $.

For a ring $ R $, we use $ T_n(R) $ to denote the $ n\times n $ upper triangular matrix ring over $ R $.

Corollary 2.10 Let $ R $ be a commutative local domain with $ J(R) = 2R $ and $ R/J(R)\cong \mathbb{Z}_2 $. The following are equivalent for $ A\in M_2(R) $:

$ (1) $ $ A $ is strongly quasi-nil clean.

$ (2) $ $ A\in (M_2(R))^{\rm qnil} $ or $ I_2-A\in (M_2(R))^{\rm qnil} $ or there exists some $ u\in U(R) $ such that $ ({\rm tr}A)^2-4{\rm det}A = u^2 $.

Proof If $ A $ and $ I_2-A $ are all not in $ (M_2(R))^{\rm qnil} $, then $ {\rm tr}A\in U(R) $ and $ {\rm det}A\in J(R) $. Since $ R/J(R)\cong \mathbb{Z}_2 $, $ U(R) = -1+J(R) $. By [8, Theorem 2.13], $ x^2-({\rm tr}A)x+{\rm det}A = 0 $ is solvable in $ R $ if and only if $ ({\rm tr}A)^2-4{\rm det}A = u^2 $ for some $ u\in U(R) $. By Theorem 2.6, we complete the proof.

It follows from Corollary 2.7.

Corollary 2.11 Let $ R $ be a commutative local ring and $ A = \left( \begin{smallmatrix} a_{11} & a_{12} \\ 0 & a_{22} \\ \end{smallmatrix} \right)\in M_2(R) $. The following statements are equivalent:

$ (1) $ $ A $ is strongly quasi-nil clean.

$ (2) $ One of the following holds:

$ (i) $ $ a_{11}, a_{22}\in 1+J(R) $ or $ a_{11}, a_{22}\in J(R) $.

$ (ii) $ One of $ a_{11} $ and $ a_{22} $ is in $ J(R) $, and the other is in $ 1+J(R) $.

Let $ R $ be a commutative local ring. By Corollary 2.11, the matrix $ A = \left( \begin{smallmatrix} j_1 & x \\ 0 & j_2 \\ \end{smallmatrix} \right)\in M_2(R) $ where $ j_1, j_2\in J(R) $ and $ x\in R $ is strongly quasi-nil clean.

For $ a\in R $, we recall that an element $ a\in R $ is said to be perfectly J-clean [3] if there exists $ e^2 = e\in {\rm comm}^2(a) $ such that $ a-e\in J(R) $. A ring $ R $ is called perfectly J-clean if every element in $ R $ is perfectly J-clean. We use $ l_a $ and $ r_a $ to denote the abelian group endomorphisms of $ R $ given by left and right multiplication by $ a $, respectively. Following [9], a local ring $ R $ is called bleached if $ l_u-r_j $ and $ l_j-r_u $ are both surjective for any $ j\in J(R) $ and any $ u\in U(R) $. Commutative local rings are well-known examples of bleached local rings. In [10, Example 4.4], the author demonstrated that over a bleached local ring $ R $, $ (T_2(R))^{\rm qnil} = J(T_2(R)) $.

Corollary 2.12 If $ R $ is a bleached local ring. Then $ A\in T_2(R) $ is strongly quasi-nil clean if and only if $ A $ is perfectly J-clean.

proposition 2.13 Let $ R $ be a ring. The following statements are equivalent:

$ (1) $ $ R $ is local and strongly quasi-nil clean.

$ (2) $ $ R $ is perfectly J-clean with the only idempotents $ 0 $ and $ 1 $.

Proof (1) $ \Rightarrow $ (2) Clearly, $ 0 $ and $ 1 $ are the only idempotents of $ R $. Since $ R $ is local, $ R $ is quasipolar and $ R^{\rm qnil} = J(R) $. Whence $ R $ is perfectly J-clean.

(2) $ \Rightarrow $ (1) Since $ R $ is perfectly J-clean, $ R/J(R) $ is boolean by [3, Theorem 4.1]. It follows that $ R/J(R)\cong \mathbb{Z}_2 $ from the only trivial idempotent $ 0 $ and $ 1 $ of $ R $. Thus $ R $ is local.