query: Isomorphism between the ring $R$ and the quotient ring $R/\langle a\rangle$
positive_passages.docid: 39576
positive_passages.text: Quotient ring being isomorphic to the initial ring
negative_passages.docid: 146028
negative_passages.text: $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
negative_passages.docid: 123130
negative_passages.text: Showing an isomorphism between exterior products of a free module
negative_passages.docid: 149856
negative_passages.text: Image of ring homomorphism is a subring
negative_passages.docid: 61292
negative_passages.text: How to know which elements are in the Quotient Ring $\mathbb{Z}[x]/\langle x, 2\rangle$?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 109596
negative_passages.text: Isomorphism from $\mathrm{Aut}(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$ to $S_3$
negative_passages.docid: 99567
negative_passages.text: Proving that an inverse ring homomorphism of an ideal is an ideal?
negative_passages.docid: 98818
negative_passages.text: Quotient ring of a localization of a ring
negative_passages.docid: 93017
negative_passages.text: Isomorphism between $\Bbb R$ and $\Bbb R(X)$?
negative_passages.docid: 70872
negative_passages.text: Isomorphism of quotient modules implies isomorphism of submodules?
negative_passages.docid: 58135
negative_passages.text: Quotient group $SL(2,\mathbb{R})/SO(2,\mathbb{R})$
negative_passages.docid: 47086
negative_passages.text: Quotient rings and free modules
negative_passages.docid: 45627
negative_passages.text: Find a sub-ring of M(2, $\mathbb{Q}$) which is isomorphic to $\mathbb{Q}$
negative_passages.docid: 85752
negative_passages.text: Show that $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_4$ as $\mathbb{Z}$-modules
negative_passages.docid: 1686
negative_passages.text: Show that quotient rings are not isomorphic
negative_passages.docid: 121449
negative_passages.text: An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$