A ring $S$ is said to be a normalizing extension of a subring $R$ if $R$ and $S$ have the same unity element and if $S=\sum_1^nRa_i$ with $Ra_i=a_iR$ for each $i$. The main result of this paper is Theorem 8: If $S$ is a semiprime normalizing extension of $R$ which is left ``torsionfree'' (i.e.\ $Is=0$ with $I$ an essential two-sided ideal of $R$ and $s$ in $S$ implies $s=0$), then (i) $R$ is semiprime, (ii) $R$ is left Goldie if and only if $S$ is left Goldie, and (iii) if $R$ and $S$ are left Goldie, then (a) regular elements of $R$ are regular in $S$, (b) $S$ has the left Ore condition with respect to the set $C$ of regular elements of $R$, and (c) the localization $C^{-1}S$ is the full quotient ring of $S$, and is a normalizing extension of $C^{-1}R$. \par \{Reviewer's remarks: This paper is related to earlier works in the literature. See, for example, a paper by \n C. Lanski\en [Proc. Amer. Math. Soc. {\bf 79} (1980), no. 4, 515--519; MR0572292 (81h:16026)] and a more recent series of papers by the reviewer and \n J. C. Robson\en [J. Algebra {\bf 72} (1981), no. 1, 237--268; MR0634625 (83f:16004); ibid. {\bf 76} (1982), no. 2, 459--470; MR0661865 (83i:16030); Trans. Amer. Math. Soc. {\bf 282} (1984), no. 2, 645--667; MR0732112 (85h:16030); J. Algebra {\bf 91} (1984), no. 1, 142--165; MR0765776 (86c:16001)]. The author invokes some of the results in these papers and points out that some other results obtained inter alia can also be found there. In terms of what is done in Theorem 8, (i) was previously known, (ii) and (iii) (a) are readily deduced from more general results of the reviewer and \n Robson\en, but (iii) (b) and (c) are new.\}