  The aim of the papers is to describe the left regular left quotient ring
${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following
algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where
$\mS_1=K\langle x,y\, | \, yx=1\rangle$, $\CI_n=K\langle \der_1, \ldots,
\der_n,\int_1,\ldots, \int_n\rangle$ is the algebra of scalar
integro-differential operators and the Jacobian algebra $\mA_1=K\langle x,\der,
(\der x)^{-1}\rangle$. The sets of left and right regular elements of the
algebras $\mS_1$, $\CI_1$, $\mA_1$ and $\mI_1=K\langle x, \der,\int\rangle$. A
progress is made on the following conjecture, \cite{Clas-lreg-quot}:
$${}'Q(\mI_n)\simeq Q(A_n)\;\; {\rm
  where}\;\; \mI_n =K\bigg\langle x_1,\ldots , x_n, \der_1, \ldots,
\der_n,\int_1,\ldots, \int_n\bigg\rangle$$ is the algebra of polynomial
integro-differential operators and $Q(A_n)$ is the classical quotient ring (of
fractions) of the $n$'th Weyl algebra $A_n$, i.e. a criterion is given when the
isomorphism holds. We produce several general constructions of left Ore and
left denominator sets that appear naturally in applications and are of
independent interest and use them to produce explicit left denominator sets
that give the localization ring isomorphic to ${}'Q(\mS_n)$ or ${}'Q(\mI_n)$ or
${}'Q(\mA_n)$ where $\mA_n:=\mA_1^{\t n}$. Several characterizations of
one-sided regular elements of a ring are given in module-theoretic and
one-sided-ideal-theoretic way.
