Strong primeness in matrix rings

The bound of uniform strong primeness of the ring Mn(R)Mn(R) of n by n matrices over the unitary ring R   is denoted mn(R)mn(R). The concepts of uniform, right and left strong primeness for matrix rings are re-interpreted in terms of bilinear equations and multiplication of vectors. These interpretations are used to prove new results. Bounds of strong primeness of unitary rings R   are linked to the bounds for Mn(R)Mn(R). The bound m2(D)m2(D) is investigated for division rings D  . Results by van den Berg (1998) and Beidar and Wisbauer (2004) linking uniform strong primeness to the existence of certain, possibly nonassociative, division algebras are generalised from fields to division rings. The result mn(D)≤2n−1mn(D)≤2n−1 of van den Berg (1998) for division rings is extended to mnn′(R)≤(2n−1)mn′(R)mnn′(R)≤(2n−1)mn′(R) for general unitary rings. In the case of formally real fields F  , it is improved to mn(F)≤2n−2mn(F)≤2n−2 for integers n>1n>1 and mn(F)≤2n−4mn(F)≤2n−4 for even n>2n>2. This improvement, used in conjunction with a generalisation of an algebraic–topological proof of Hopf's theorem on real division algebras, yields m2k+1(R)=m2k+2(R)=2k+1m2k+1(R)=m2k+2(R)=2k+1. Bounds on mn(R)mn(R) for other n are also obtained.