Basic Commutative Algebra I

It is the mark of the educated mind to use for each subject the degree of exactitude which it admits. — Aristotle

Commutative algebra is the basic of many other topics in mathematics. The understanding of quite a portion of commutative algebra is needed as foundation for the understanding of topics such as algebraic geometry, which is essentially the study of the geometry of polynomial rings, number theory which I know nothing about, or the invariant theory which is a little out-fashioned nowadays.

I assume the reader is familiar with the definition of commutative rings, which can be found easily on the internet. Two textbooks are strongly recommended for complete beginners,

• A First Course in Abstract Algebra by John Fraleigh,
• An Introductory Course in Commutative Algebra by A. W. Chatters and C. R. Hajanavis and the all-time most classic ,
• Introduction to Commutative Algebra by M.F.Atiyah and I.G.Macdonald.

Starting with some examples of commutative rings,

• The ring of integers $\mathbb{Z}$, for plus, minus and multiplication is defined but not division.
• The Gaussian integer $\mathbb{Z}[i]$, it is the set of complex integers of form $m+in$ where $m,n$ are integers and $i^2=-1$.

We will assume the rings to be commutative and has multiplication identity element denoted by $1$. The identity element for addition is denoted by $0$.

Note that if $0=1$, then for all elements $x$ in a ring $A$, we have

$$\begin{array}{}\text{(1)}& x=x_1=x_0=0\end{array}$$

thus A is a ring with only $0$, namely $A$ is the zero ring.

We are interested in maps that preserves multiplication, addition and identity between rings, and give them a name: ring homomorphism, homo- comes from Greek “homos” meaning “the same”. To be specific, a ring homomorphism should satisfy

• $f(x+y)=f(x)+f(y)$
• $f(xy)=f(x)f(y)$
• $f(1)=1$

A subring of $A$ is a subset of $A$ which is also a ring, namely closed under addition, subtraction, multiplication and has identity.

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The most important concept in the study of rings is ideal. It is sometimes denoted by a German letter, such as $\mathfrak{a}\mathfrak{,}\mathfrak{b}\mathfrak{,}\mathfrak{g}$, etc.

Definition. An ideal $\mathfrak{a}$ of a ring $A$ is a subset of $A$ which is an additive subgroup and is such that $A\mathfrak{a}\subseteq \mathfrak{a}$ (i.e., $x\in A,y\in \mathfrak{a}$ implies $xy\in \mathfrak{a}$).

An ideal is like the virus in a ring, infects everting it touches, whatever it multiplies is absorbed into the ring itself. For example, in the ring of integers, if number 3 is in an ideal, then all the numbers of form $3x$ is also in the same ideal.

Definition. The quotient ring is $A/\mathfrak{a}$. The multiplication of the quotient ring is inherited from $A$. The elements in $A/\mathfrak{a}$ are the cosets of $\mathfrak{a}$ in $A$.

By taking the quotient $A/\mathfrak{a}$, we are essentially treating all the elements in $\mathfrak{a}$ as zeros. Take the ring of integers for example, let $3\mathbb{Z}$ be the ideal generated by $3$, namely the set of numbers of form $3x,x\in \mathbb{Z}$ then the quotient ring $\mathbb{Z}/3\mathbb{Z}$ means that, in the original $\mathbb{Z}$, we treat all elements in $3\mathbb{Z}$ as zeros, i.e. numbers such as $3,6,-3,-6,\dots$ are all effectives zero in $\mathbb{Z}/3\mathbb{Z}$. What does it mean to “treat 3 in $\mathbb{Z}$ as zero”? It means that 3 multiplies anything is zero, and 3 adds anything is the same thing. Plus, if $x-y=3$ then $x=y$ in the quotient ring since $3$ is seen as zero. There are only three distinct elements in this quotient ring then, denoted by $[0],[1],[2]$.

If an ideals $\mathfrak{a}$ is a virus in a ring $A$ and “infection” is “multiplication”, then the quotient ring $A/\mathfrak{a}$ is what we get after killing the virus.

Proposition. There is a one-to-one order-preserving, correspondence between the ideals $\mathfrak{b}$ of $A$ which contain $\mathfrak{a}$, and the ideal $\bar{\mathfrak{b}}$ of $A/\mathfrak{a}$, given by $\mathfrak{b}\mathfrak{=}{\varphi }^{\mathfrak{-}\mathfrak{1}}\mathfrak{(}\bar{\mathfrak{b}}\mathfrak{)}$, where the ring homomorphism $\varphi :A\to A/\mathfrak{a}$ maps each $x$ in $A$ to its coset $x+\mathfrak{a}$.

In the above proposition, the order is given by set inclusion, for example $A\subset B$ defines an order between $A$ and $B$. To understand the correspondences consider the simples example, $\mathfrak{a}$ itself. $\mathfrak{a}$ is an ideal in $A$ and $\varphi (\mathfrak{a}\mathfrak{)}$ is an ideal in $A/\mathfrak{a}$, the zero ideal. Conversely, the pre-image of zero ideal in $A/\mathfrak{a}$, i.e., ${\varphi }^{-1}(0)$ is $x\in A\mid \varphi (x)=0=\mathfrak{a}$ by construction. There is a one-to-one correspondence between $\mathfrak{a}$ as an ideal in $A$ and $0$ as the zero ideal in $A/\mathfrak{a}$. Given a bigger ideal $\mathfrak{b}$ which contains $\mathfrak{a}$, we can “cure” $\mathfrak{b}$ of virus $\mathfrak{a}$ and get another ideal, but in $A/\mathfrak{a}$, and the converse also holds.

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Preview of class 2

Definition.

• A Domain is a nonzero ring with in which $ab=0$ implies $a=0$ or $b=0$ or both. If $ab=0$ and $a,b\ne 0$, then $a,b$ are called zero devisors, so a domain can be said to be a non-zero ring without zero devisors. It makes a lot of operations familiar to us from the manipulation of integer numbers still valid. For example $\mathbb{Z}$ is a Domain.
• An integral domain is just a commutative domain.
• A unit of a ring is an invertible element for the multiplication of the ring. That is, an element $u$ of a ring is a unit if there exists $v$ in the same ring such that $uv=vu=1$.
• A prime element is an element which is not zero or a unit, such that $p|ab$ implies $p|a$ or $p|b$, where $p|a$ means $p$ divides $a$.