The behavior of ascending chain conditions on submodules of bounded finite generation in direct sums

We construct a ring R which has the ascending chain condition on n-generated right ideals for each n ≥ 1 (also called the right pan-acc property), such that no full matrix ring over R has the ascending chain condition on cyclic right ideals. Thus, the right pan-acc property is not a Morita invariant. Moreover, a direct sum of (free) modules with pan-acc does not necessarily even have 1-acc.