A Remark on Restricted Serial Rings 1

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 35 (1--2), (1980), 49--51.

A REMARK ON RESTRICTED SERIAL RINGS 1

By A. WIDIGER (Halle)

1. In [1], IWANAGA posed the problem whether there exists a restricted uni- serial ring which is neither left nor right noetherian. In this note we give an example for such a ring. Moreover we show that we can drop the assumption of Proposition 2 of [1] that A is a noetherian ring. Thus we give a characterization of non-prime, non-noetherian restricted serial rings (for its definition see below) even without identity. This is essentially based on the main result of [3]. A (unitary) module is called serial if the lattice of submodules with respect to inclusion is linearly ordered. A ring with identity is called left serial (right serial) if it is a direct sum of serial left (right) modules and a left-right serial ring is said to be serial. As in [1] we call a ring A restricted serial provided that every proper homomorphic image of A is a left and right artinian serial ring. J(A) denotes the Jacobson radical of the ring A, A, the n• matrix ring over A. 2. By symmetry we can restrict ourselves to the case where A is not right noetherian. THEOREM. Let A be a non-prime non right noetherian ring. Then A is restricted serial if and only if A is precisely one of the following BI (I) A=]~)" -o|, S an infinite division ring, B asimple unitary S,-left module. L (II) A is a full matrix ring over a completely primary ring R, J(R)2=(0). J(R) has infinite dimension as R/J(R)-right module, and either (a) J(R) is, on each side, a direct sum of a minimal two-sided ideal of R and a simple one-sided ideal or (b) J(R) is a minimal two-sided ideal of R (III) A= Km, M a unitary Sn-Km-bimodule, S, K division rings, M has infinite dimension as Kin-right module, and either (a) J(A) is, on each side, a direct sum of a minimal two-sided ideal of A and a simple one-sided ideal or (b) J(A) is a minimal two-sided ideal of A. PROOF. The "if" part is almost trivial. To prove the "only if" part we show first that A is not right artinian. If it were then, since it is not right noetherian, it

1 The paper was written during a study leave in Budapest.

Aeta I~Iathema~ica Academiae Scient~arum Hungar~cae 35, 1980 50 A. WIDIGER would contain a Priifer subgroup Z(p ~) ([2], Satz 10.10). Then clearly there exists a proper factorring of A without identity since Z(p ~) is in the annihilator of A ([2], Satz 10.3) contrary to our assumption. Thus A is not right artinian but every proper factor ring is right artinian. Since A cannot be nilpotent, A is of type (II), (IID or (IV) of Satz 5 of [3]. If in case (II) or (III) of the theorem, J(R) or M is a simple bimodule, it is clearly a minimal ideal of A. Assume that J(R) is not a minimal ideal of R in (!I). Let B be an ideal of R, B~J(R). Then J(R)=BOK=BOL with right or left ideals K or L, resp., since J(R) is a completely reducible module. Since R/B is serial, K is a simple right ideal and L is a simple left ideal of R. Moreover, if C is another ideal of R such that (O)~C~J(R), then so is C~ B. If C 0 B= (0), then B ~ B/BAC_~ (B+C)/C would be a right artinian A-module since A/C is right artinian. Since A/B is itself an artinian A-module, A would be right artinian contradicting our first statement. Therefore CV1B#(O). Again B=Cf~B~L1 with right ideal L1 and R/(B~C) is not serial if L1r Thus LI=(0), CNB=B and similarly C(~B=C. Hence B is a minimal ideal. The same procedure is applicable to did and the theorem is proved. 3. EXAMPLE. Let Q be the field of rational numbers, x transcendental over Q and consider the chain 4

Q = Q(r c Q(t/ =..c Q(2 = .... Define K= ~) Q(22-"). n=l

Then K is a field, in fact an infinite dimensional algebraic extension of Q. Clearly x is transcendental over K too, K(x) is an infinite dimensional algebraic extension of Q(x). Moreover, there is a chain 4 Q(x) of subfields of K(x) with K(x) = ~J Q (x) (22-"). Let

Clearly A is neither left nor right noetherian. To prove that

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