Constructing division rings as module-theoretic direct limits

Constructing Division Rings as Module-theoretic Direct Limits

If R is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring D generated by a homomorphic image of R is a rule putting on all free R-modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how D may be constru...

Direct Division in Factor Rings

Conventional techniques for division in the polynomial factor ring F[z]/〈m〉 or the integer ring Zn use a combination of inversion and multiplication. We present a new algorithm that computes the division directly and therefore eliminates the multiplication step. The algorithm requires 2 degree(m) (resp. 2 log 2 n) steps, each of which uses only shift and multiply-subtract operations.

Tensors as module homomorphisms over group rings

Braman [1] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars. This extends the familiar notion that matrices form the set of all linear transformations over vectors with real-valued scalars. This result is based upon a circulant-based tensor multiplication due to Kilmer et al. [4]. In this work,...

Triangulaizability of Algebras Over Division Rings

Generalized Covering Groups and Direct Limits