A ring \(R\) is called right 1-inversive, if every left regular element has a right inverse and fully right inversive if every left regular matrix, square or not, has a right inverse. The author shows that any ring can be embedded in a fully right inversive ring \(S\) such that all left regular matrices over the original ring become right invertible over \(S\). In proving this result the author first shows that for a left regular element \(c\) the natural homomorphism \(R\to R\langle c'\mid cc' = 1\rangle\) is an embedding and then applies it to matrix rings. Further, it is shown that every fully right inversive right semihereditary ring \(R\) with ACC on direct summands of \(R_ R\) is semisimple.