In this paper we study a condition right FGTF on a ring R, namely when all finitely generated torsionless right R-modules embed in a free module. We show that for a von Neuman regular (VNR) ring R the condition is equivalent to every matrix ring Rn is a Baer ring; and this is right-left symmetric. Furthermore, for any Utumi VNR, this can be strengthened: R is FGTF iff R is self-injective.

Let $R$ be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in $R$ is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.

We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E\left(R\right)$ and a $u\in U\left(R\right)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann\left(a^{+}\right)$ and $u\in U\left(R\right)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR⟹aR\cong bR$.

It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\{e_1,\cdots ,e_n\}$ of idempotents such that all $e_{i}R{e}_i$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.