The Jacobson radical of the Laurent series ring.
All rings in this paper are associative with nonzero identity element. A ring \(A\) is called a ring of bounded index, if there exists a positive integer \(n\) such that \(a^n=0\) for every nilpotent element \(a\in A\). A module is said to be finite-dimensional if it does not contain an infinite direct sum of nonzero submodules. A right finite-dimensional ring with the maximum condition on right annihilators is called a right Goldie ring. For a ring \(A\), the intersection of all prime ideals of \(A\) is called the prime radical of \(A\). A ring \(A\) is said to be semiprime if \(A\) does not have nonzero nilpotent ideals. The intersection of all maximal submodules of the module \(M\) is denoted by \(J(M)\) and is called the Jacobson radical of the module \(M\). A module \(M\) is said to be semiprimitive if \(J(M)=0\). For a ring \(A\) and an automorphism \(\varphi\) of \(A\), the author considers the skew Laurent series ring \(A(\!(x,\varphi)\!)\) with coefficient ring \(A\) formed by all series \(f=\sum_{i=k}^{+\infty}f_ix^i\), where \(x\) is a variable, \(k\) is an integer (possibly negative), and all the coefficients \(f_i\) are contained in \(A\). In the ring \(A(\!(x,\varphi)\!)\), addition is naturally defined and multiplication is defined by the formula \(xa=\varphi(a)x\) for all \(a\in A\). So, if \(\varphi=1_A\), we obtain the ordinary Laurent series ring \(A(\!(x)\!)\). For every subset \(N\) of \(A\), \(N(\!(x,\varphi)\!)\) denotes the subset of the ring \(A(\!(x,\varphi)\!)\) consisting of series \(f=\sum_{i=k}^{+\infty}f_ix^i\) whose coefficients \(f_i\) are contained in \(N\). The author shows that if \(A\) is a ring such that the prime radical \(N\) of \(A\) is nilpotent and the factor ring \(A/N\) is a right Goldie ring (for example, this is the case if \(A\) is a right Noetherian ring), then for every automorphism \(\varphi\) of the ring \(A\), the Jacobson radical of the skew Laurent series ring \(A(\!(x,\varphi)\!)\) is nilpotent and it coincides with \(N(\!(x,\varphi)\!)\). In particular, if \(A\) is a semiprime right Goldie ring, then for every automorphism \(\varphi\) of the ring \(A\), the skew Laurent series ring \(A(\!(x,\varphi)\!)\) is semiprimitive. Moreover, the author shows that if \(A\) is a ring with prime radical \(N\) and the factor ring \(A/N\) is a ring of bounded index, then the Jacobson radical of the Laurent series ring \(A(\!(x)\!)\) coincides with \(N(\!(x)\!)\). In particular, if \(A\) is a semiprime ring of bounded index, then the Laurent series ring \(A(\!(x)\!)\) is semiprimitive.
scientific article; zbMATH DE number 5358152