Definition of Rings

Rings

Definition 1 A monoid object in the symmetric monoidal category \((\Ab,\otimes, \mathbb{Z})\) is called a ring. (§Abelian Groups, ⁋Definition 11)

Following the convention for abelian groups, the operation of \(A\) is denoted by \(+\). In this context, the multiplication \(\mu:A\otimes A \rightarrow A\) of a ring \(A\) is denoted by \(\cdot\), and when there is no ambiguity, \(\alpha\cdot \beta\) is abbreviated as \(\alpha\beta\). From

\[\Hom_\Ab(A\otimes A,A)\cong\Bilin(A,A;A)\]

we see that \(\mu\) is bilinear. That is,

\[(\alpha+\beta)\gamma=\alpha\gamma+\beta\gamma,\quad \alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma\]

hold. Also, \(\mu\) is associative, and

\[\eta:\mathbb{Z}\rightarrow A\]

determines the (multiplicative) identity element \(1\) of \(A\) through the image of \(1\in \mathbb{Z}\). Thus, a set \(A\) being a ring means that there exist two binary operations \(+,\cdot\) and elements \(0,1\) satisfying the following conditions.

1. \((A, +, 0)\) is an abelian group.
2. \((A,\cdot,1)\) is a monoid.
3. Distributivity holds between \(\cdot\) and \(+\).

Depending on the author, the existence of a multiplicative identity element may not be included in the definition of a ring, but we call such cases pseudo-rings, or rng (ring without the “i”). In any case, we will mostly work with commutative rings (with unity).

Proposition 2 For any elements \(\alpha,\beta\) of a ring \(A\), the following hold.

1. \(\alpha0=0\alpha=0\),
2. \(\alpha(-\beta)=(-\alpha)\beta=-(\alpha\beta)\).

Proof

1. Since \(0\) is the additive identity, from the equation

   \[0\alpha=(0+0)\alpha=0\alpha+0\alpha\]

   we obtain \(0\alpha=0\). Similarly, we obtain \(\alpha0=0\).

2. By result 1,

   \[0=\alpha0=\alpha(\beta+(-\beta))=\alpha\beta+\alpha(-\beta)\]

   and therefore \(-(\alpha\beta)=\alpha(-\beta)\). Similarly, \((-\alpha)\beta=-(\alpha\beta)\) is obtained.

In general, in Definition 1, we do not assume \(0\neq 1\), but if \(0=1\), then by item 1 of Proposition 2, for any \(\alpha\in A\),

\[\alpha=\alpha\cdot 1=\alpha\cdot 0=0\]

holds, so \(A=\{0\}\). Such a ring is called the zero ring.

Ring Homomorphisms

Definition 3 For two rings \(A,B\), a function \(\phi:A \rightarrow B\) is a ring homomorphism if for any \(x,y\),

\[\phi(\alpha+\beta)=\phi(\alpha)+\phi(\beta),\quad \phi(\alpha\beta)=\phi(\alpha)\phi(\beta),\quad \phi(1)=1\]

hold.

Taking these as morphisms, we define

• The category \(\Ring\) of rings,
• The category \(\Rng\) of rngs,
• The category \(\cRing\) of commutative rings

and so on. In \(\Ring\) and \(\cRing\), \(\mathbb{Z}\) is an initial object and \(\{0\}\) is a terminal object, while in \(\Rng\), \(\{0\}\) is a zero object.

By definition, a ring homomorphism \(\phi:A\rightarrow B\) is also a group homomorphism between the abelian groups \((A,+,0)\) and \((B,+,0)\). The kernel \(\ker f\) of a ring homomorphism \(\phi\) is defined as the kernel of this group homomorphism. That is, \(\ker \phi=\phi^{-1}(0)\). By §Homomorphisms, ⁋Proposition 4, \(\phi\) is injective if and only if \(\ker \phi=\{0\}\).

Free Ring on an Abelian Group

By definition, a ring is an abelian group, and a ring homomorphism is also a homomorphism between abelian groups. Thus there exists a forgetful functor \(U:\Ring \rightarrow \Ab\), which simply forgets the multiplicative structure. This functor has a right adjoint, and its left adjoint \(F: \Ab \rightarrow \Ring\) is given by the graded abelian group

\[F(G)=\bigoplus_{n\geq 0} G^{\otimes n}\]

Here, an element \(\alpha_n\) of \(G^{\otimes n}\) can be written in the form

\[\alpha_n=\sum_{i\in I} \alpha^i_{n1}\otimes\cdots\otimes \alpha^i_{nn},\qquad \text{$\alpha_{nj}^i\in A$, $I$ finite}\]

and elements of \(F(G)\) are of the form

\[(\alpha_0,\alpha_1,\ldots )=(\alpha_0,0,\ldots)+(0,\alpha_1,\ldots)+\cdots,\qquad \text{$\alpha_n=0$ for all but finitely many $n$}\]

But since each \(\alpha_n\) can be identified by how many elements’ tensor product its components \(\alpha^i_{n1}\otimes\cdots \alpha^i_{nn}\) consist of, i.e., which \(G^{\otimes n}\) it belongs to, by abusing notation we can write \((0,\ldots, 0, \alpha_n,0,\ldots)\) as \(\alpha_n\), and then any element of \(F(G)\) can be written as

\[\sum_{i\in I} \alpha_{i1}\otimes \cdots\otimes \alpha_{in_i}\]

Now we need to define multiplication on \(F(G)\). If multiplication is well-defined, by distributivity,

\[\left(\sum_{i\in I} \alpha_{i1}\otimes \cdots\otimes \alpha_{in_i}\right)\left(\sum_{j\in J} \beta_{j1}\otimes \cdots\otimes \beta_{jn_j}\right)=\sum_{(i,j)\in I\times J}(\alpha_{i1}\otimes\cdots \otimes \alpha_{in_i})(\beta_{j1}\otimes\cdots\otimes \beta_{jn_j})\]

must hold. Conversely, if we define only the products of \(\alpha_{i1}\otimes\cdots \otimes \alpha_{in_i}\) and \(\beta_{j1}\otimes\cdots\otimes \beta_{jn_j}\), then the multiplication of all elements of \(F(G)\) is defined through the above equation. And we define

\[(\alpha_{i1}\otimes\cdots \otimes \alpha_{in_i})(\beta_{j1}\otimes\cdots\otimes \beta_{jn_j})=\alpha_{i1}\otimes\cdots\otimes \alpha_{in_i}\otimes \beta_{j1}\otimes\cdots \beta_{jn_j}\]

By the coherence theorem, this defines a ring structure on \(F(G)\), with the additive identity being \(0=(0,0,\ldots)\) and the multiplicative identity being \(1=(1,0,\ldots)\). The functoriality of \(G\mapsto F(G)\) can be easily proved, and moreover, the following holds.

Proposition 4 For \(F\) defined above and the forgetful functor \(U:\Ring \rightarrow \Ab\), we have \(F\dashv U\).

Proof

Let an arbitrary ring \(A\) and an abelian group \(G\) be given. We need to prove

\[\Hom_\Ring(F(G), A)\cong \Hom_\Ab(G, U(A))\]

For any ring homomorphism \(\phi: F(G) \rightarrow A\), composing with the inclusion \(i:G\hookrightarrow F(G)\) yields an abelian group homomorphism \(\phi\circ i:G \rightarrow U(A)\).

Conversely, given an abelian group homomorphism \(f:G \rightarrow U(A)\), the equation

\[\sum_{i\in I} \alpha_{i1}\otimes \cdots\otimes \alpha_{in_i}\mapsto \sum_{i\in I} f(\alpha_{i1})\otimes \cdots\otimes f(\alpha_{in_i})\]

defines a ring homomorphism \(\phi:F(G) \rightarrow A\).

We can verify that the two functions \(\Hom_\Ring(F(G), A) \rightarrow\Hom_\Ab(G, U(A))\) and \(\Hom_\Ab(G, U(A))\rightarrow \Hom_\Ring(F(G), A)\) defined in this way are inverses of each other, and that this bijection is natural.

Subrings and Ideals

Definition 5 A subset \(S\) of a ring \((A,+,-,\cdot,0,1)\) is a subring if \((S,+,-,\cdot,0,1)\) has a ring structure.

On the other hand, the following holds.

Proposition 6 For any ring homomorphism \(\phi:A \rightarrow B\), \(\ker \phi\) is a subring of \(A\).

Proof

We have already verified that \(\ker \phi\) is a subgroup of the abelian group \((A,+,0)\), so it suffices to show that \(\ker \phi\) is closed under multiplication. For any \(\alpha,\beta\in\ker \phi\),

\[\phi(\alpha\beta)=\phi(\alpha)\phi(\beta)=0\cdot 0=0\]

so \(\alpha\beta\in\ker \phi\) holds.

Looking carefully at the above proof, we can see that even if only one of the two elements \(\alpha,\beta\) belongs to \(\ker \phi\), \(\alpha\beta\) still belongs to \(\ker \phi\). This is defined as follows.

Definition 7 Let a ring \(A\) be given. A subset \(\mathfrak{a}\subseteq A\) is a left ideal (resp. right ideal) if \(\mathfrak{a}\) is a subgroup of \((A,+,0)\) and for any \(x\in\mathfrak{a}\) and \(\alpha\in A\), \(\alpha x\in\mathfrak{a}\) (resp. \(x\alpha\in\mathfrak{a}\)) holds.

If \(\mathfrak{a}\) is both a left ideal and a right ideal, it is called a two-sided ideal.

For convenience, we will only prove statements about left ideals (or two-sided ideals), but all statements about left ideals also hold for right ideals with appropriate modifications. Since most rings we will actually use are commutative, the distinction between left ideals, right ideals, and two-sided ideals is unnecessary.

It is easy to prove that the intersection of left ideals is a left ideal. On the other hand, for any element \(x\) of \(A\), the set

\[Ax=\{\alpha x\mid\alpha\in A\}\]

is a subgroup under addition, and for any \(\beta\in A\) and \(\alpha x\in Ax\), \(\beta(\alpha x)=(\beta\alpha)x\in Ax\), so \(Ax\) is a left ideal of \(A\). These can be shown to coincide with the smallest left ideal containing \(x\), i.e., the intersection of all left ideals containing \(x\), and the same argument applies to right ideals and two-sided ideals.

More generally, defining the sum \(\mathfrak{a}+\mathfrak{b}\) of left ideals of \(A\) as the set

\[\mathfrak{a}+\mathfrak{b}=\{x+y\mid x\in \mathfrak{a},y\in \mathfrak{b}\}\]

we see that \(\mathfrak{a}+\mathfrak{b}\) is again a left ideal, and in fact this is the smallest left ideal containing both \(\mathfrak{a}\) and \(\mathfrak{b}\). For any elements \(x_1,\ldots, x_n\) of \(A\), the ideal

\[Ax_1+\cdots+Ax_n\]

is the smallest among all left ideals containing \(x_1,\ldots, x_n\). Similarly, we can define

\[x_1A+\cdots+x_nA,\qquad Ax_1A+\cdots +Ax_nA\]

and these are the smallest right ideal and two-sided ideal containing \(x_1,\ldots, x_n\), respectively. If \(A\) is commutative, the notions of left ideal, right ideal, and two-sided ideal all coincide, so these are collectively denoted by \((x_1,\ldots, x_n)\).

On the other hand, for any left ideal \(\mathfrak{a}\), \(1\in\mathfrak{a}\) is equivalent to \(\mathfrak{a}=A\). Therefore, for \(\mathfrak{a}\subsetneq A\), we must have \(1\not\in\mathfrak{a}\).

Definition 8 For a ring \(A\) and an ideal \(\mathfrak{m}\), if there exists no ideal \(\mathfrak{a}\) satisfying \(\mathfrak{m}\subsetneq\mathfrak{a}\subsetneq A\), then \(\mathfrak{m}\) is called a maximal ideal.

Let any ideal \(\mathfrak{a}\) of \(A\) be given. Consider the collection of ideals of \(A\) containing \(\mathfrak{a}\) but different from \(A\) itself. This collection is an inductive set, so it has a maximal element. ([Set Theory] §Axiom of Choice, ⁋Theorem 4) It is easy to verify that this maximal element is a maximal ideal. Thus we obtain the following.

Theorem 9 (Krull) For an ideal \(\mathfrak{a}\subsetneq A\) of a ring \(A\), there always exists a maximal ideal \(\mathfrak{m}\) of \(A\) containing \(\mathfrak{a}\).

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References

[Bou] Bourbaki, N. Algebra I. Elements of Mathematics. Springer. 1998.

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