Quaternion rings over local rings

Abstract

In 1843, Hamilton (1805–1865) discovered the 4-dimensional division algebra [Formula: see text] over the field [Formula: see text] of real numbers. Hamilton’s big discovery is the following beautiful multiplications for the basis [Formula: see text] :[Formula: see text] [Formula: see text] contains the field [Formula: see text] of complex numbers; therefore [Formula: see text]. Frobenius (1849–1917) showed the following outstanding theorem. Theorem (Frobenius). Up to isomorphism, the only finite-dimensional non-commutative division algebra over [Formula: see text] is [Formula: see text]. Starting from given any ring [Formula: see text] and a free right [Formula: see text]-module [Formula: see text], the quaternion ring [Formula: see text] is canonically defined by the multiplications [Formula: see text]. For a commutative field [Formula: see text] with [Formula: see text], [Formula: see text] is a division ring or isomorphic to the ring of [Formula: see text] matrices over [Formula: see text]. This is a classical theorem. For nonzero [Formula: see text] in the center of a ring [Formula: see text], the generalized quaternion ring [Formula: see text] is defined. [Formula: see text] is [Formula: see text]. In [I. Kikumasa, K. Koike and K. Oshiro, Complex rings and quaternion rings, East-West J. Math. 21 (2019) 1–19; I. Kikumasa, G. Lee and K. Oshiro, Complex Rings, Quaternion Rings and Octonion Rings (Lambert Academic Publishing, 2020); G. Lee and K. Oshiro, Quaternion rings and octonion rings, Front. Math. China 12(1) (2017) 143–155], quaternion rings and generalized quaternion rings over division rings or other rings are studied. Now, in this paper, from ring theoretic viewpoints, we study quaternion rings [Formula: see text] and generalized quaternion rings [Formula: see text] over local rings [Formula: see text]. From our results, we can clearly look over several classical results on [Formula: see text] and [Formula: see text] over commutative fields [Formula: see text].