If $I,J$ are index sets, $R$ a commutative unital ring, $\mathfrak{a},\mathfrak{b}$ ideals of polynomial rings $R[x_i; i\!\in\!I]$, $R[y_j; j\!\in\!J]$, and $\langle\langle\ldots\rangle\rangle$ is the ideal generated by $\ldots$, is there an isomorphism of $R$-algebras
$R[x_i; i\!\in\!I]/\mathfrak{a} \:\oplus\: R[y_j; j\!\in\!J]/\mathfrak{b} \;\cong\; R[x_i, y_j; i\!\in\!I, j\!\in\!J]/\langle\langle\mathfrak{a},\mathfrak{b},x_iy_j; i\!\in\!I,j\!\in\!J\rangle\rangle?$
The map $(f(x)\!+\!\mathfrak{a},\,g(y)\!+\!\mathfrak{b})\longmapsto f(x)\!+\!g(y)\!+\!\langle\langle\ldots\rangle\rangle$ is not unital.
If $\cong$ does not hold, what other generators of the ideal $\langle\langle\ldots\rangle\rangle$ must I take?
Note: from what I understand, there is an isomorphism of $R$-algebras
$R[x_i; i\!\in\!I]/\mathfrak{a} \:\otimes\: R[y_j; j\!\in\!J]/\mathfrak{b} \;\cong\; R[x_i, y_j; i\!\in\!I, j\!\in\!J]/\langle\langle\mathfrak{a},\mathfrak{b}\rangle\rangle.$