By K.P.S. Bhaskara Rao

K.P.S. B.R. conception of Generalized Inverses Over Commutative earrings (CRC Press Inc, 2002)(ISBN 0203218876)

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Proof. Let m ∈ H(e). Then, we have for some u, u′ , v, v ′ ∈ M e = mu , m = eu′ , e = vm , m = v′ e . Therefore em = e(eu′ ) = eu′ = m and in the same way me = m. This shows that m ∈ eM e. Since m(eue) = mue = e , (eve)m = evm = e , 1032 1033 1034 the element m is both right and left invertible in M . Hence, m belongs to the group of units of eM e. Conversely, if m ∈ eM e is right and left invertible, we have mu = vm = e for some u, v ∈ eM e. Since m = em = me, we obtain mHe. J. Berstel, D. Perrin and C.

If P = Q, we say that it is a K-relation over Q. The set of all K-relations between P and Q is denoted by K P ×Q . Let m ∈ K P ×Q be a K-relation between P and Q. For p ∈ P , the row of index p of m is denoted by mp∗ . It is the element of K Q defined by (mp∗ )q = mpq . Similarly, the column of index q of m is denoted by m∗q . It is an element of K P . Let P, Q, R be three sets and let K be a complete semiring. For m ∈ K P ×Q and n ∈ K Q×R , the product mn is defined as the following element of K P ×R .

Proof. Set N (z) = I − M z, where I is the identity matrix and z is a variable. The polynomial N (z) can be considered both as a polynomial with coefficients in the ring of m × m-matrices or as an m × m-matrix with coefficients in the ring of real polynomials in the variable z. The polynomial N (z) is invertible in both structures, and its inverse N (z)−1 = (I − M z)−1 can in turn be viewed as a power series with coefficients in the ring of m × m-matrices or as a matrix whose coefficients are rational fractions in the variable z.

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