A NONHOMOLOGICAL PROOF OF SEMIPERFECTNESS IN MATRIX RINGS

A. Y. M. Chin

Introduction. Let R be an associative ring with unit. An element e of R is said to be an idempotent if e^{2} = e. Two idempotents e, f of R are said to be orthogonal if ef = fe = 0. A nonzero idempotent e of R is said to be primitive if it cannot be written as the sum of two nonzero orthogonal idempotents. If e is an idempotent of R such that eRe is a local ring, that is eRe has exactly one maximal ideal, then e is said to be local. It is known (see [2] for example) that every local idempotent is primitive. However, the converse is not necessarily true. For example, 1 is a primitive but not local idempotent of ℤ. For an ideal I of R, we say that idempotents of R/I can be lifted to R if for every idempotent u + I ∈ R/I, there exists an idempotent e^{2} = e ∈ R such that e − u ∈ I.

Denote the Jacobson radical of R by J(R) and the ring of n × n matrices over R by M_{n}(R). R is said to be semiperfect if R/J(R) is Artinian and idempotents of R/J(R) can be lifted to R. It has been shown by Kaye [1] via the Morita Duality Theorem that R is semiperfect if and only if M_{n}(R) is semiperfect. The purpose of this paper is to give a nonhomological proof of this result.

All rings considered in this paper are assumed to be associative with unit.

1. Some Preliminaries. We first state the following result by B. J. Mueller [3] on conditions on a ring which are equivalent to being semiperfect.

\underline{Theorem 1.1}. Let R be a ring. The following conditions are equivalent:

(i) R is semiperfect; (ii) Every primitive idempotent of R is local and there is no infinite set of orthog-

onal idempotents in R; (iii) The unit 1 ∈ R is the finite sum of some orthogonal local idempotents.

In what follows, for i, j = 1, …, n, we let E_{ij} = (e_{rs}) denote the n × n matrix over R such that

e_{rs} = \{ \begin{matrix}1 & if (r, s) = (i, j) \\ 0 & if (r, s) ≠ (i, j) \end{matrix} , r, s = 1, …, n.

\underline{Proposition 1.2}. Let R be a ring. If e is a primitive idempotent of R, then eE_{tt} is a primitive idempotent of M_{n}(R) for t = 1, …, n.

\underline{Proof}. Clearly, eE_{tt} is an idempotent of M_{n}(R). Suppose that eE_{tt} is not primitive. Then we may write

eE_{tt} = X + Y

for some nonzero orthogonal idempotents X = (x_{ij}), Y = (y_{ij}) ∈ M_{n}(R). We next note the following properties:

(i) e = x_{tt} + y_{tt}; (ii) x_{ij} = −y_{ij} for all i, j, (i, j) ≠ (t, t); (iii) X^{2} = X ⇒ ∑_{k=1}^{n} x_{ik}x_{kj} = x_{ij} for all i, j; (iv) Y^{2} = Y ⇒ ∑_{k=1}^{n} y_{ik}y_{kj} = y_{ij} for all i, j;

(v) XY = 0 ⇒ ∑_{k=1}^{n} x_{ik}y_{kj} = 0 for all i, j; (vi) Y X = 0 ⇒ ∑_{k=1}^{n} y_{ik}x_{kj} = 0 for all i, j. We now show that x_{ij} = 0 for all (i, j) ≠ (t, t). In order to do this, we shall show that (I) x_{ij} = 0 for all i = 1, …, n; j = 1, …, t − 1, t + 1, …, n, and (II) x_{ij} = 0 for all i ≠ t, j = t. We first show (I). Let j ∈ {1, …, t − 1, t + 1, …, n}. Then (k, j) ≠ (t, t) for any k = 1, …, n. By (ii) we have that x_{kj} = −y_{kj} for every k = 1, …, n. It follows that

x_{ij} = ∑_{k=1}^{n} x_{ik}x_{kj} (by (iii))

= ∑_{k=1}^{n} x_{ik}(−y_{kj})

= − ∑_{k=1}^{n} x_{ik}y_{kj} = 0 (by (v))

for every i = 1, …, n. Hence, (I) is proven. Next we shall show (II). Let i ∈ {1, …, t − 1, t + 1, …, n}. Then (i, k) ≠ (t, t) and it follows from (ii) that x_{ik} = −y_{ik} for every k = 1, …, n. Therefore,

x_{it} = ∑_{k=1}^{n} x_{ik}x_{kt} (by (iii))

= ∑_{k=1}^{n}(−y_{ik})x_{kt}

= − ∑_{k=1}^{n} y_{ik}x_{kt} = 0 (by (vi)).

Hence, (II) is proven. Thus, we have shown that x_{ij} = 0 for all (i, j) ≠ (t, t). By (ii), it then follows that y_{ij} = 0 for all (i, j) ≠ (t, t).

Now by (iii),

x_{tt}^{2} + ∑_{\begin{matrix}k=1 \\ k≠t \end{matrix}}^{n} x_{tk}x_{kt} = x_{tt}.

But since x_{tk} = 0 for all k ≠ t, we have that x_{tt}^{2} = x_{tt}. Next, from (v) we have

x_{tt}y_{tt} + ∑_{\begin{matrix}k=1 \\ k≠t \end{matrix}}^{n} x_{tk}y_{kt} = 0.

But since x_{tk} = 0 for all k ≠ t, we have that x_{tt}y_{tt} = 0. Note that since y_{ij} = 0 for all (i, j) ≠ (t, t), it can be similarly shown that y_{tt}^{2} = y_{tt} and y_{tt}x_{tt} = 0. Also, note that x_{tt} ≠ 0 and y_{tt} ≠ 0. Indeed, if x_{tt} = 0, then X = (x_{ij}) = 0; a contradiction. Similarly, we would have a contradiction if y_{tt} = 0. Thus, we have shown that e = x_{tt} + y_{tt} where 0 ≠ x_{tt} = x_{tt}^{2}, 0 ≠ y_{tt} = y_{tt}^{2} and x_{tt}y_{tt} = y_{tt}x_{tt} = 0. This contradicts the fact that e is a primitive idempotent of R. Hence, eE_{tt} must be a primitive idempotent of M_{n}(R).

\underline{Lemma 1.3}. Let R be a ring and let e be a nonzero idempotent of R. Then (eE_{tt})M_{n}(R)(eE_{tt}) ≅ eRe for every t = 1, …, n.

\underline{Proof}. Define θ: (eE_{tt})M_{n}(R)(eE_{tt}) → eRe as follows:
θ((eE_{tt})(x_{ij})(eE_{tt})) = ex_{tt}e, (x_{ij}) ∈ M_{n}(R).

By routine verification, θ is a ring isomorphism.

2. The Main Result. The main result in this paper has been obtained by Kaye [1] and is as follows:

\underline{Theorem 2.1}. A ring R is semiperfect if and only if M_{n}(R) is semiperfect.

We give here a different proof of this result using the preliminaries in the preceding section.

\underline{Proof of Theorem 2.1}. Suppose that R is semiperfect. By condition (iii) of Theorem 1.1, there exists a finite set of orthogonal local idempotents {e_{1}, …, e_{m}} in R such that

1 = e_{1} + ⋯ + e_{m}.

Let I_{n} denote the identity matrix of M_{n}(R). Clearly

I_{n} = e_{1}E_{11} + ⋯ + e_{m}E_{11}
+ e_{1}E_{22} + ⋯ + e_{m}E_{22} + …
+ e_{1}E_{nn} + ⋯ + e_{m}E_{nn}.

We note that (e_{i}E_{tt})^{2} = e_{i}E_{tt} and (e_{i}E_{tt})(e_{j}E_{ss}) = 0 if i ≠ j or s ≠ t where i, j = 1, …, m; s, t = 1, …, n. We now show that each e_{i}E_{tt} is local. Since e_{i}Re_{i} ≅ (e_{i}E_{tt})M_{n}(R)(e_{i}E_{tt}) (by Lemma 1.3) and e_{i}Re_{i} is local, so is (e_{i}E_{tt})M_{n}(R)(e_{i}E_{tt}). It follows that each e_{i}E_{tt} is local. Thus, we have shown that the identity element of M_{n}(R) is the finite sum of some orthogonal local idempotents. Hence, by condition (iii) of Theorem 1.1, M_{n}(R) is semiperfect.

Now suppose that M_{n}(R) is semiperfect. Note that if e_{i} and e_{j} are orthogonal idempotents of R, then e_{i}E_{11} and e_{j}E_{11} are orthogonal idempotents of M_{n}(R). Then since there is no infinite set of orthogonal idempotents in M_{n}(R) (by (ii) of Theorem 1.1), it follows that there is also no infinite set of orthogonal idempotents in R.

Next we show that every primitive idempotent of R is local. Let e be a primitive idempotent of R. From Proposition 1.2 we have that eE_{11} is a primitive idempotent of M_{n}(R). Since M_{n}(R) is semiperfect, so eE_{11} is local and therefore, (eE_{11})M_{n}(R)(eE_{11}) is a local ring. From Lemma 1.3 we have that eRe ≅ (eE_{11})M_{n}(R)(eE_{11}) is local. Therefore e is a local idempotent. Thus, we have shown that R satisfies condition (ii) of Theorem 1.1. Hence, R is semiperfect.

\underline{References}

1. S. Kaye, “Ring Theoretic Properties of Matrix Rings,” Canad. Math. Bull., 10 (1967), 365–374.

2. J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, Mass., 1966.

3. B. J. Mueller, “On Semi–Perfect Rings,” Illinois J. Math., 14 (1970), 464–467.

A. Y. M. Chin Institute of Mathematical Sciences Faculty of Science University of Malaya 50603 Kuala Lumpur email: acym@omega.math.um.edu.my