Annihilators in rings with involution

We study semiprime superalgebras with superinvolution whose symmetric elements are not zero divisors, and semiprime superalgebras with superinvolution, with nonzero odd part, whose skewsymmetric elements are not zero divisors. We prove that, in both cases, such superalgebras are a domain or the subdirect sum of a domain and its opposite.

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

In this paper, a new finite element method, termed the generalized modal element method (GMEM), is proposed. In GMEM, the element stiffness is derived by decomposing element deformation patterns into individual element generalized modes, where different methods are used to construct the generalized modes. Specifically, three different modal construction methods, including analytical method, assumed displacement method and traditional finite element technique, are proposed for developing the element generalized modes. The concept of modal local coordinate systems is also proposed to ensure the element frame invariance when using polynomial displacement functions, which successfully enables one to use the analytical solutions derived from governing differential equations to develop high accuracy element formulations. An asymmetric hexahedral solid element and a symmetric hexahedral solid element are subsequently derived by using GMEM. The displacement functions of the elemental 24 generalized modes are expressed in terms of Cartesian coordinates so that the element behavior is independent of mesh distortions. Furthermore, the first 21 generalized modes are derived from analytical method making the element capable of avoiding common locking phenomena. Several benchmark problems are performed to demonstrate the accuracy and performance of the new element formulations in linear static and frequency analysis.