Editing Isomorphism theorems (section)

== Rings ==
The statements of the theorems for [[ring (mathematics)|rings]] are similar, with the notion of a normal subgroup replaced by the notion of an [[ideal (ring theory)|ideal]].

=== Theorem A (rings) ===
Let <math>R</math> and <math>S</math> be rings, and let <math>\varphi:R\rightarrow S</math> be a [[ring homomorphism]]. Then:
# The [[Kernel (algebra)#Ring homomorphisms|kernel]] of <math>\varphi</math> is an ideal of <math>R</math>,
# The [[image (mathematics)|image]] of <math>\varphi</math> is a [[subring]] of <math>S</math>, and
# The image of <math>\varphi</math> is [[ring isomorphism|isomorphic]] to the [[quotient ring]] <math>R/\ker\varphi</math>.
In particular, if <math>\varphi</math> is surjective then <math>S</math> is isomorphic to <math>R/\ker\varphi</math>.<ref>{{Cite web |last=Moy |first=Samuel |date=2022 |title=An Introduction to the Theory of Field Extensions |url=https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Moy.pdf |access-date=Dec 20, 2022 |website=UChicago Department of Math}}</ref>

=== Theorem B (rings) ===
Let <math>R</math> be a ring.  Let <math>S</math> be a subring of <math>R</math>, and let <math>I</math> be an ideal of <math>R</math>.  Then:
# The [[Ideal (ring theory)#Ideal operations|sum]] <math>S+I=\{s+i\mid s\in S,i\in I\}</math> is a subring of <math>R</math>,
# The intersection <math>S\cap I</math> is an ideal of <math>S</math>, and
# The quotient rings <math>(S+I)/I</math> and <math>S/(S\cap I)</math> are isomorphic.

=== Theorem C (rings)===
Let ''R'' be a ring, and ''I'' an ideal of ''R''.  Then
# If <math>A</math> is a subring of <math>R</math> such that <math>I \subseteq A \subseteq R</math>, then <math>A/I</math> is a subring of <math>R/I</math>.
# Every subring of <math>R/I</math> is of the form <math>A/I</math> for some subring <math>A</math> of <math>R</math> such that <math>I \subseteq A \subseteq R</math>.
# If <math>J</math> is an ideal of <math>R</math> such that <math>I \subseteq J \subseteq R</math>, then <math>J/I</math> is an ideal of <math>R/I</math>.
# Every ideal of <math>R/I</math> is of the form <math>J/I</math> for some ideal <math>J</math> of <math>R</math> such that <math>I \subseteq J \subseteq R</math>.
# If <math>J</math> is an ideal of <math>R</math> such that <math>I \subseteq J \subseteq R</math>, then the quotient ring <math>(R/I)/(J/I)</math> is isomorphic to <math>R/J</math>.

=== Theorem D (rings)===
Let <math>I</math> be an ideal of <math>R</math>. The correspondence <math>A\leftrightarrow A/I</math> is an [[subset|inclusion]]-preserving [[bijection]] between the set of subrings <math>A</math> of <math>R</math> that contain <math>I</math> and the set of subrings of <math>R/I</math>. Furthermore, <math>A</math> (a subring containing <math>I</math>) is an ideal of <math>R</math> [[if and only if]] <math>A/I</math> is an ideal of <math>R/I</math>.<ref>{{cite book | last1=Dummit | first1=David S. | first2=Richard M. | last2=Foote |  title=Abstract algebra | url=https://archive.org/details/abstractalgebra00dumm_304 | url-access=limited | location=Hoboken, NJ | publisher=Wiley | date=2004 | page=[https://archive.org/details/abstractalgebra00dumm_304/page/n259 246] | isbn=978-0-471-43334-7}}</ref>