Ideals, Quotient Rings and the Isomorphism Theorems for Rings

Given: A ring $R$ and a subset $I\subseteq R$.

A two-sided ideal is an additive subgroup closed under left and right multiplication by ring elements (Left, right and two-sided ideals).

A nonempty subset closed under $x{y}^{-1}$ is a subgroup (One-step subgroup test: a nonempty $H\subseteq G$ is a subgroup iff $g{h}^{-1}\in H$ for all $g,h\in H$; the identity and the inverses of $H$ are then those of $G$).

A ring has additive inverses and distributive multiplication (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

Given: A commutative ring $R$ and a subset $S\subseteq R$.

$(S)$ is the intersection of ideals containing $S$ (The ideal generated by a subset and principal ideals).

The ideal criterion uses subtraction and absorption (Ideal criteria and intersections of ideals).

Multiplication in a commutative ring commutes (Commutative ring).

Given: Two-sided ideals $I,J⊴R$.

$I+J$ and $IJ$ are the indicated elementwise and finite-sum sets (The sum $I+J$ and product $IJ$ of two-sided ideals).

The ideal criterion is subtraction closure plus two-sided absorption (Ideal criteria and intersections of ideals).

Ring multiplication distributes over finite sums (In any ring $0\cdot a=a\cdot 0=0$, $(-a)b=a(-b)=-(ab)$, $(-a)(-b)=ab$, $(-1)a=-a$ and $a(b-c)=ab-ac$).

Given: A ring homomorphism $f:R\to S$.

A ring homomorphism preserves addition, multiplication, $0$, and $1$ (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

Ring homomorphisms preserve additive inverses (A ring homomorphism satisfies $f(0)=0$, $f(-a)=-f(a)$ and $f(ma)=mf(a)$ for $m\in \mathbb{Z}$, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms).

The group kernel is the inverse image of $0$ in the additive groups (The kernel and image of a group homomorphism).

A two-sided ideal is an additive subgroup with two-sided absorption (Left, right and two-sided ideals).

Given: A ring $R$ and an additive subgroup $J\le (R,+)$.

The displayed rule is the proposed quotient multiplication (The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$).

A two-sided ideal is an additive subgroup absorbing multiplication on both sides (Left, right and two-sided ideals).

Ring multiplication is distributive (In any ring $0\cdot a=a\cdot 0=0$, $(-a)b=a(-b)=-(ab)$, $(-a)(-b)=ab$, $(-1)a=-a$ and $a(b-c)=ab-ac$).

Coset equality is membership of a difference in the subgroup ($x\in aH$ iff $a^{-1}x\in H$, and $aH=bH$ iff $a^{-1}b\in H$).

Additive cosets are available for an additive subgroup (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$).

Given: A ring $R$ and a two-sided ideal $I⊴R$.

$R\mathrm{/}I$ has the stated coset addition and multiplication (The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$).

This multiplication is well defined for a two-sided ideal (Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal).

The additive cosets already form a group (For $N⊴G$, the cosets form a group with identity $N$ and inverse $(gN{)}^{-1}=g^{-1}N$).

Associativity and distributivity hold in $R$ (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

Given: A ring $R$ and a two-sided ideal $I⊴R$.

$R\mathrm{/}I$ is a ring with the stated coset operations (For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$).

A ring homomorphism preserves addition, multiplication, and identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

The additive quotient map is surjective (The canonical projection $\pi :G\to G\mathrm{/}N$, $\pi (g)=gN$, is a surjective group homomorphism).

A kernel is the inverse image of the identity element (The kernel and image of a group homomorphism).

The coset-equality criterion gives $r+I=0+I$ exactly when $-r\in I$, hence exactly when $r\in I$ because $I$ is an additive subgroup ($x\in aH$ iff $a^{-1}x\in H$, and $aH=bH$ iff $a^{-1}b\in H$).

Given: A natural number $n$, viewed as a nonnegative integer.

The canonical quotient map is a surjective ring homomorphism (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

The additive quotient $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$ is $(\mathbb{Z}\mathrm{/}n,+)$ (For every $n\in \mathbb{N}$, the congruence-class group $(\mathbb{Z}\mathrm{/}n,+)$ is the quotient group $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$).

Congruence classes carry the published modular ring operations (For every natural $n$, $(\mathbb{Z}\mathrm{/}n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

$[a{]}_n+[b{]}_n=[a+b{]}_n$ and $[a{]}_n[b{]}_n=[ab{]}_n$ (Addition and multiplication on $\mathbb{Z}\mathrm{/}n$ by $[a{]}_n+[b{]}_n=[a+b{]}_n$ and $[a{]}_n[b{]}_n=[ab{]}_n$).

$\mathbb{Z}$ is a commutative ring with identity (The integers form a commutative ring).

Given: A two-sided ideal $I⊴R$ and a ring homomorphism $f:R\to S$ with $I\subseteq \ker f$.

The additive group of a ring is abelian (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

A two-sided ideal is an additive subgroup (Left, right and two-sided ideals).

Every subgroup of an abelian group is normal (Every subgroup of an abelian group is normal).

The canonical quotient map is a surjective ring homomorphism (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

A group homomorphism killing a normal subgroup factors uniquely through its quotient (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

A ring homomorphism preserves addition, multiplication, and identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

Given: A ring homomorphism $f:R\to S$.

A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring (A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).

$\ker f$ is an ideal (The kernel of a ring homomorphism is a two-sided ideal).

Ring homomorphisms preserve additive inverses and products (A ring homomorphism satisfies $f(0)=0$, $f(-a)=-f(a)$ and $f(ma)=mf(a)$ for $m\in \mathbb{Z}$, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms).

The underlying group map is isomorphic modulo its kernel to its image (First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$).

A ring homomorphism preserves $1$ (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

Given: A unital subring $S\subseteq R$ and a two-sided ideal $I⊴R$.

A subring contains the ambient identity and is closed under ring operations (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication).

The subring criterion tests subtraction and multiplication (Subring criterion: $S\subseteq R$ is a subring if and only if $1_R\in S$ and $a-b\in S$ and $ab\in S$ for all $a,b\in S$; and an intersection of subrings is a subring).

An ideal is an additive subgroup with two-sided absorption (Left, right and two-sided ideals).

Intersections of ideals and the ideal criterion are valid (Ideal criteria and intersections of ideals).

Given: A unital subring $S\subseteq R$ and a two-sided ideal $I⊴R$.

$S+I$ is a subring, $I⊴S+I$, and $S\cap I⊴S$ (If $S$ is a subring and $I$ is an ideal of $R$, then $S+I$ is a subring, $I$ is an ideal of $S+I$, and $S\cap I$ is an ideal of $S$).

The first ring isomorphism theorem identifies a ring modulo a kernel with its image (First isomorphism theorem for rings: $R\mathrm{/}\ker f\cong \mathrm{im}f$).

The canonical quotient map is a surjective ring homomorphism (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

Given: Two-sided ideals $I\subseteq J⊴R$.

Ideals are additive subgroups with absorption (Left, right and two-sided ideals).

$R\mathrm{/}I$ is a ring of additive cosets (For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$).

The quotient map has kernel $I$ (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

Given: Two-sided ideals $I\subseteq J⊴R$.

$J\mathrm{/}I$ is an ideal of $R\mathrm{/}I$ (If $I\subseteq J$ are ideals of $R$, then $J\mathrm{/}I$ is an ideal of $R\mathrm{/}I$).

A ring modulo a kernel is isomorphic to the image (First isomorphism theorem for rings: $R\mathrm{/}\ker f\cong \mathrm{im}f$).

Quotient projections are surjective ring homomorphisms (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

Given: An ideal $I⊴R$ and $\pi :R\to R\mathrm{/}I$.

$\pi$ is a surjective ring homomorphism with kernel $I$ (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

Kernels of ring homomorphisms are ideals (The kernel of a ring homomorphism is a two-sided ideal).

Ideals are additive subgroups with absorption (Left, right and two-sided ideals).

$R\mathrm{/}I$ is a ring (For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$).

Given: A commutative ring $R$ and a two-sided ideal $P⊴R$.

A prime ideal is proper and satisfies $ab\in P\Rightarrow a\in P$ or $b\in P$ (Prime ideals and maximal ideals in a commutative ring).

Products in $R\mathrm{/}P$ are $(a+P)(b+P)=ab+P$ (For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$).

An integral domain is a nonzero commutative ring without zero divisors (Zero divisor, and integral domain: a commutative ring with $1\mathrm{\ne }0$ and no zero divisors).

The canonical projection $R\to R\mathrm{/}P$ has kernel $P$; equivalently, $a+P=0+P$ exactly when $a\in P$ (The canonical projection $R\to R\mathrm{/}I$ is a surjective ring homomorphism with kernel $I$).

Given: A commutative ring $R$ and a proper ideal $M⊴R$.

A maximal ideal has no proper intermediate ideal (Prime ideals and maximal ideals in a commutative ring).

$R\mathrm{/}M$ is a quotient ring with its usual coset operations (For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$).

A field is a commutative ring in which every nonzero element is invertible (Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring).

The definition of field requires a multiplicative inverse for each nonzero element (Field).

The ideal criterion verifies ideals by subtraction and absorption (Ideal criteria and intersections of ideals).

Given: A maximal ideal $M$ of a commutative ring $R$.

$R\mathrm{/}M$ is a field when $M$ is maximal ($R\mathrm{/}M$ is a field if and only if $M$ is a maximal ideal).

$R\mathrm{/}P$ is a domain exactly when $P$ is prime ($R\mathrm{/}P$ is an integral domain if and only if $P$ is a prime ideal).

Every field is an integral domain (Every field is a commutative ring with $1\mathrm{\ne }0$; it is an integral domain, and it is a commutative division ring).

Given: A nonzero commutative ring $R$ and a proper ideal $I⊴R$.

A maximal ideal is a maximal proper ideal under inclusion (Prime ideals and maximal ideals in a commutative ring).

Ideals are additive subgroups with multiplication absorption (Left, right and two-sided ideals).

An ideal criterion and intersection closure are available (Ideal criteria and intersections of ideals).

Assuming the Axiom of Choice, a nonempty poset in which every chain has an upper bound has a maximal element (Zorn's lemma).

A chain is a subset linearly ordered by the ambient order (Chain in a poset).

A maximal element has no strictly larger element in the poset (Maximal element and greatest element).

A nonzero ring has $1\mathrm{\ne }0$ and ring operations distribute (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).