Abstract. Let R be a ring, A ¼ M n ðRÞ and : A ! A a surjective additive map preserving zero Jordan products, i.e. if x; y 2 A are such that xy þ yx ¼ 0, then ðxÞ ðyÞ þ ðyÞ ðxÞ ¼ 0. In this paper, we show that if R contains 1 2 and n 5 4, then ¼ ', where ¼ ð1Þ is a central element of A and ' : A ! A is a Jordan homomorphism.2000 Mathematics Subject Classification: 15A04, 47B49
We construct a counterexample to settle simultaneously the following questions all in the negative: (1) Is a regular subdirect product of simple artinian rings unit-regular? (2) If R is a regular ring such that every nonzero ideal of R contains a nonzero ideal of bounded index, is R unit-regular? (3) Is a regular ring with a Hausdorff family of pseudo-rank functions unit-regular? (4) If R is a regular ring which contains no infinite direct sum of nonzero pairwise isomorphic right ideals, is R unit-regular? (5) Is a regular Schur ring unit-regular?In [1] Goodearl proposed a list of open problems on regular rings. Some involve potential sufficient conditions for a regular ring to be unit-regular. The primary aim of this paper is to construct a counterexample for the questions 6, 7, 8, 9 (second part) and 11 in GoodearΓs book.Among others the sixth question asks: Is a regular subdirect product of simple artinian rings always unit-regularΊ In [4] Tyukavkin has shown that any regular algebra over an uncountable field, which is a subdirect product of countably many simple artinian rings, is unitregular. Recently, Goodearl and Menal [2] have generalized this result by showing that any regular algebra over an uncountable field, which has no uncountable direct sums of nonzero right or left ideals, must be unit-regular; in particular, any regular algebra over an uncountable field, which has a rank function, is unit-regular. In this paper we shall construct an example of a regular ring which is a subdirect product of countably many simple artinian rings but is not unit-regular.Let F be a countable field, F[t] the ring of polynomials over F in an indeterminate t, and F{t) the quotient field of F [t]. Define an exponential valuation d on F(t) by dr{t) = +oo if r{t) = 0 and dr(t) = n if r{t) = t n f(t)/g(t) where n is an integer and f(t), g(t) e F[t] with t \ f(t)g(t).Let V be the valuation ring associated with d, namely, V = {r(ή e F(t)\dr(t) > 0}. Note that F[t], F(t) and V are all countable. Consequently, V is a countable-dimensional vector space over F. Let VQ, v\ 9 ..., υ n ,... be a basis of V over F. First, we may assume that dVi Φ dVj for i φ j. Suppose that n is the least integer such that dv n = dvi for some i < n. Choose α f G F so that f«/^/ -α z G ίF; then 9(v Λ -α, v, ) > <9t>/. If <9(^« -α/V/) = <9^ for some j < n, then d(v n -α/V/ -CLJVJ) > dVj for some α y G /\ Continuing this process we get a t^ such that dv' n φ dVi for all i < n and that {vo,vu...,v n _ u υ' n } spans the same subspace as {vQ 9 υ\ 9 ... 9 v n -\ 9 v n } does. Next, we assume, by reordering, that dv 0 < dvi < dv 2 18 CHEN-LIAN CHUANG AND PJEK-HWEE LEEwith a k Φ 0, we see that dv = dv k . Since Vo 9 v\ 9 V2,... span the whole space V, we must have dvo = 0, dv\ = 1, 9^2 = 2 and so on.We begin by constructing a ring which is similar to that in Bergman's example [1; Example 4.26]. Let S be the set of those x G E = End/r(F) such that (JC -ά)t n V = 0 for some a G .F(ί) and some nonnegative integer n. As in [1; p. 47] we observe that a depends...
We give a systematic account on the relationship between a ring R with involution and its subrings S and K, which are generated by all its symmetric elements or skew elements respectively. I. Introduction. Let R be a ring with involution * and 5 the subring generated by the set S of all symmetric elements in R. The relationship between R and S has been studied by various authors. In proved that if_R is prime or semi-prime, so is 5. In §2 of this paper, we show that S can inherit a number of ring-theoretic properties such as primitivity, semisimplicity, absence of nonzero nil ideals etc.. In doing so, a notion called symmetric subring, which is a generalization of S and its *-homomorphic images, is introduced so that a group of theorems of the same type, including Lanski's results, can be proved via a more or less unified argument. We show also that numerous radicals of S are merely the contractions from those of R. As a consequence, we see that R modulo its prime radical behaves much like S in many respects.In §3 we establish a corresponding theory for K, the subring generated by all skew elements. The only result hitherto known concerning K was as follows [4], [12]: If R is_simple and dim z i? >4, then K -R L As a matter of fact, the subring K 2 is more closely related to JR than K is. We apply thejtechnique developed in §2 to study the relationship between R and K
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