Derivations of prime rings
This paper has two types of results. First, there are proved the following theorems about rings of constants of finite dimensional \(\partial\)-Lie algebras of external derivations of rings with positive characteristic: Theorem 1. Let L be a finite-dimensional \(\partial\)-Lie algebra of external derivations of the prime ring R with positive characteristic. Then the ring of constants \(R^ L\) is a left Goldie ring if and only if R is a left Goldie ring. Theorem 2. Let R be a prime ring. Then \(Q_ M(R)^ L=Q_ M(R^ L)\), where \(Q_ M(B)\) denotes the two-sided Martindale quotient ring of B and R,L are as in theorem 1. Similar results for rings of invariants of finite groups were obtained by \textit{V. K. Kharchenko} [Algebra Logika 16, 313-363 (1977; Zbl 0397.16037)]. Theorem 1 answers Question 2 from \textit{V. K. Kharchenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 435-462 (1981; Zbl 0469.16019)]. Secondly the author gives complete answers to the following questions which were formulated by \textit{I. N. Herstein} [(1): Can. Math. Bull. 21, 369-370 (1978; Zbl 0412.16018)]: (2) J. Algebra 60, 567-574 (1979; Zbl 0436.16014)]: 1) Let R be a prime ring and R satisfies an identity \(S_ n(d(x_ 1),...,d(x_ n))=0,\) where d is a non zero derivation on R. Has R the identity \(S_ n(x_ 1,...,x_ n)=0?\) 2) Let R be a prime ring, d be a derivation on R such that \((d(x))^ n=0\) for an integer \(n\geq 1\). Holds \(d=0?\) Question 1 has answer ''yes'', if \(n=2\) [\textit{I. N. Herstein}, (1)], and ''no'', if \(n\geq 9\) [\textit{A. Kovacs}, Can. Math. Bull. 22, 339-344 (1979; Zbl 0436.16015)]. Question 2 has answer ''yes'', if d is an inner derivation. In the paper under review it is proved that the ring of infinite row- finite matrices satisfies the identity \(S_ 3(d(x_ 1),d(x_ 2),d(x_ 3))\), where d is an inner derivation, defined by the matrix unit \(e_{12}\), yet it is well-known, that this ring not satisfies any identities. Furthermore, there is proved Proposition 3. Let R be a prime ring and d be a derivation on R, which is externel on \(Q_ M(R)\). Assume that R satisfies the differential identity \(S_ n(d(x_ 1),...,d(x_ n))=0,\) then R satisfies the identity \(S_ n(x_ 1,...,x_ n).\) The following theorem extends the above quoted result of Herstein and gives a complete answer to question 2: Theorem 4. Let R be a prime ring and d be a derivation on R, furthermore, \((d(x))^ n=0\) for all \(x\in R\). Then \(d=0\).
finite-dimensional \(\partial\)- Lie algebra
scientific article; zbMATH DE number 3853315