query: If there is an automorphism between elements in a ring, why does this mean that the quotient ring over certain ideals are isomorphic?
positive_passages.docid: 73235
positive_passages.text: Rings $R/I \cong h(R)/h(I)\,$ for injective ring hom $h$
negative_passages.docid: 78053
negative_passages.text: There are at least three mutually non-isomorphic rings with $4$ elements?
negative_passages.docid: 642
negative_passages.text: Does a surjective ring homomorphism have to be surjective on the unit groups?
negative_passages.docid: 99567
negative_passages.text: Proving that an inverse ring homomorphism of an ideal is an ideal?
negative_passages.docid: 41629
negative_passages.text: Structure of ideals in the product of two rings
negative_passages.docid: 22604
negative_passages.text: Can $R \times R$ be isomorphic to $R$ as rings?
negative_passages.docid: 6646
negative_passages.text: A (non-artificial) example of a ring without maximal ideals
negative_passages.docid: 47086
negative_passages.text: Quotient rings and free modules
negative_passages.docid: 8587
negative_passages.text: What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?
negative_passages.docid: 57732
negative_passages.text: Rings of fractions and homomorphisms
negative_passages.docid: 34990
negative_passages.text: Ring homomorphism defined on a field
negative_passages.docid: 110984
negative_passages.text: Ideals and prime ideals in a commutative ring.
negative_passages.docid: 91294
negative_passages.text: order of quotient ring
negative_passages.docid: 78118
negative_passages.text: Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
negative_passages.docid: 3778
negative_passages.text: If the matrix rings over two Rings of the same size are isomorphic, then the scalar rings are isomorphic
negative_passages.docid: 117536
negative_passages.text: Why is the localization at a prime ideal a local ring?
negative_passages.docid: 64025
negative_passages.text: Ring homomorphisms from $\Bbb Q$ into a ring