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a b s t r a c t Keywords: Strongly prime Uniformly strongly prime Matrix ringThe bound of uniform strong primeness of the ring M n (R)of n by n matrices over the unitary ring R is denoted m n (R). The concepts of uniform, right and left strong primeness for matrix rings are reinterpreted 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). The bound m 2 (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 m n (D) ≤ 2n − 1of van den Berg (1998) for division rings is extended to m nn (R) ≤ (2n − 1)m n (R)for general unitary rings. In the case of formally real fields F , it is improved to m n (F ) ≤ 2n − 2 for integers n > 1and m n (F ) ≤ 2n − 4for even n > 2. This improvement, used in conjunction with a generalisation of an algebraic-topological proof of Hopf's theorem on real division algebras, yields m 2 k +1 (R) = m 2 k +2 (R) =2 k+1 . Bounds on m n (R)for other n are also obtained.

An s-cap n-flat, or an n-dimensional cap of size s, is a pair (S, F ) where F is an n-dimensional affine space over Z/3Z and the size-s subset S of F contains no triple of collinear points. The cap set problem in dimension n asks for the largest s for which an s-cap n-flat exists. This series of articles investigates the cap set problem in dimensions up to and including 7. This is the second paper in the series.By applying and adapting methods from the first paper in the series, we systematically classify all 5-dimensional caps of size at least 41.