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Let Vn(Rn) be the universal ring with respect to em- beddings of the matrix ring Rn into n X n matrix rings over commutative rings. A construction and a representation is given for this ring. As a main tool in the construction, it is proved that every R homomorphism of Rn9 R a com- mutative ring, is the restriction of an inner automorphism of Un, for some U 2 R. Using this, a necessary and sufficient condition for n2 matrices in Rn to be matrix units is given. 1* Introduction and notationsAll rings to be considered in this paper, except those denoted specifically as matrix rings, will be commutative rings with unit. All homomorphisms are unitary. The unit of a subring coincides with the unit of its over-ring. Denote by Rn the ring of n x n matrices over a ring R. Let ψ R —• S be a ring homomorphism then η induces a homomorphism ηn:Rn-*Sn given by: ηn{ri3) = (^(r^ )). If AeRn,(A)iS will denote the (i, j)th entry of A. The identity element and the standard matrix units of all matrix rings will be denoted by I and {-E^} respectively. Let A be an R algebra. It was proved by Amitsur ((1), Theorem 2) that there exists a commutative R algebra V%(A), and a map p: A— > (V%(A))m which is universal for homomorphisms of A into m x m matrix rings over commutative rings, i.e.; (1) For every τ: A —> Hm9 with H a commutative R algebra, there exists a homomorphism η: VZ(A) —• H such that the following diagram is commutative

In this paper Goldie-like conditions are put on a Jordan matrix ring J = H ( R n , γ a ) J = H({R_n},{\gamma _a}) which are necessary and sufficient for R to be a ∗ \ast -prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.

Necessary and sufficient conditions are obtained for a ring lying in a third-order matrix algebra over the field of p-adic numbers to have a finite number of nonisomorphic indecomposable representations.

It is well-known that the ring Qn of nxn matrices over a lattice-ordered ring Q may be lattice-ordered by prescribing that a matrix is positive exactly when each of its entries is positive. We conjecture in case Q is the field of rational numbers that this is essentially the only lattice-order of the matrix ring in which the multiplicative identity 1 is positive and settle the conjecture in case n=2. There are however other lattice-orders of Q2 in which 1 is not positive. A complete description of this family is obtained.

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

If R is any ring then the set of isomorphism types of finitely generated projective right R-modules may be made into a semigroup by defining.

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that(1) U is a progenerator in the category of left R-modules, and(2) S ≅ (EndR U)opp as rings.

where the ei1 are the usual unit matrices. For example, we could select n left ideals Al, * * *, An of either F or a subring of F and then let Fij=Aj, i, j=1, . I n. If F is a (skew) field and the Fij satisfying (1) are all nonzero, then R defined by (2) is easily shown to be a prime ring. The main result of this paper (1.3) is that if F is a right ring of quotients of Fil then (F)X is a right ring of quotients of R and there exists a subring K of F and a nonzero diagonal matrix dCR such that (K)n is a subring of dRd-1 and F is a ring of quotients of K. This result is used to give new proofs of the Faith-Utumi theorem [2] and of Goldie's theorem [l].

Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ring R together with a mapping, X —> ϕ(X), of R into a lattice A with 0 and 1 satisfyingis called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" and A is a 'Valuation lattice." If R is a lattice-valued ring and a mapping d is defined by

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.

The fact that every unitary matrix can be expressed in the form A-1A∗ (observed by C. R. Johnson) is interpreted as a "Hilbert Theorem 90" in rings with an involution. Several observations connected with this are included.

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