invariance of defining sets for abelian codes by isomorphism

Question

An abelian code is an ideal of the $\mathbb{F}$ algebra: $$ A(n_1,\ldots,n_l):= \dfrac{\mathbb{F}[X_1,\ldots,X_l]}{[X_{1}^{n_1}-1, \ldots,X_{l}^{n_l}-1]} $$ Given an ideal $C$ in this algebra, there exists a defining set $D(C)$ that characterizes it entirely. See: https://arxiv.org/abs/1101.1803 for more details. In the article "Information Sets From Defining Sets in Abelian Codes" by Jacobo and Bernal, available at https://arxiv.org/abs/1101.1803 page 7993, there is a statement that says the following: If $\varphi$ is an isomorphism: $$ \varphi:\mathbb{Z}_n\rightarrow\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2} $$ and $C$ is a cyclic code over $\mathbb{F}[X_1]/[X^{n}-1]$ with defining set $\mathcal{D}(C)$, then $\varphi(C)$ is an abelian code over $\mathbb{F}[X_1,X_2]/[X^{n_1}_{1}-1,X^{n_2}_{2}-1]$, and it holds that, $\mathcal{D}(\varphi(C))=\varphi(\mathcal{D}(C))$. I managed to prove this result when $\varphi$ is the isomorphism obtained by the Chinese Remainder Theorem, but not in the general case. I am grateful for any help.