### 数学代写|交换代数代写commutative algebra代考|MATH2301

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## 数学代写|交换代数代写commutative algebra代考|The radical of an ideal

One source for the concept of radical of an ideal is the simplification process of switching from the complete factorization of an integer or a polynomial to its square-free factorization in which one omits any multiplicity higher than 1 .

This crude idea underwent various stages, eventually ending up in the following formal definition: the radical of an ideal $I \subset R$ is the set $\left{a \in R \mid \exists r \geq 0, a^{r} \in I\right}$.

This is easily seen to be an ideal of $R$ containing the ideal $I$. In this book, it is denoted by the symbol $\sqrt{I}$. An ideal is said to be radical if it coincides with its radical (same vocable, twisting the grammar). Clearly, the radical of an ideal is a radical ideal.

Determining a set of generators of $\sqrt{I}$ given a set of generators of $I$ is a hard knuckle (an exception is the case of an ideal generated by monomials in a polynomial ring over a field). In order to express the radical of $I$, one needs a knowledge of other ideals related to $I$, the so-called minimal prime ideals associated to $I$. A prime ideal is an extremely relevant building part of the commutative algebra compound and will be reviewed next.
One of the nice properties of taking the radical of an ideal is the following:
$$\sqrt{I J}=\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J}$$

## 数学代写|交换代数代写commutative algebra代考|Prime and primary ideals

A prime ideal is the most notable instance of a radical ideal. In fact, the two notions are more deeply intertwined than is predicted by their formal definitions. It is not very clear who has the exact priority for the inception of this concept, with Kronecker claiming he had it before the Dedekind-Noether advances (see Historic note, Subsection 1.3.1),
Recall the formal definition.
Definition 1.1.4. An ideal $I \subset R$ is prime if it satisfies any of the following equivalent conditions:
(1) Given $a, b \in R$ such that $a b \in I$, then $a \in I$ or $b \in I$.
(2) Given ideals $J, J^{\prime} \subset R$ such that $J J^{\prime} \subset I$, then $J \subset I$ or $J^{\prime} \subset I$.
(3) The residue class ring $R / I$ has no proper zero-divisors.
Recall that the third condition above is transcribed in the notion of an integral domain. Here, one assumes that in an integral domain $1 \neq 0$ (i. e., the zero ring is not considered to be an integral domain). Likewise, for condition (3) to be equivalent to (1) and (2), one takes for granted that a prime ideal is always proper. This has the additional convenience that a prime ideal is always contained in a maximal ideal, by a suitable use of Zorn’s lemma (Kuratowski-Zorn lemma: a partially ordered set such that every totally ordered subset has an upper bound, necessarily contains at least one maximal element).

Clearly, a prime ideal is a radical ideal. In fact, just as easily one sees that the intersection of an arbitrary collection of prime ideals is a radical ideal. There is a converse to this statement.

Proposition $1.1 .5$ (Krull). The radical of an ideal is the intersection of the family of prime ideals containing it.

Proof. Let $I \subset R$ be an ideal. Clearly, $\sqrt{I}$ is contained in any prime ideal that contains $I$. Conversely, let $u \in R \backslash \sqrt{I}$ and set $S=\left{u^{n} \mid n \geq 0\right}$. By assumption, $I \cap S=\emptyset$. By a ready application of Zorn’s lemma, one can find an ideal $P \subset R$ maximal in the (nonempty) family of ideals that contain $I$ and do not intersect $S$. The proof will be completed if one shows that $P$ is a prime ideal since then $P$ will be a prime ideal containing $I$ such that $u \notin P$. Thus, let $a, b \in R$ be such that $a b \in P$, but neither $a$ nor $b$ belongs to $P$. Then the ideals $(P, a)$ and $(P, b)$ are both strictly larger than $P$, hence by the maximality assumption both intersect $S$. Let $m, n$ be suitable integers such that $u^{m} \in(P, a)$ and $u^{n} \in(P, b)$. Writing down these two conditions, multiplying them out and using the condition $a b \in P$ yields $u^{m+n} \in P$, contradicting the assumption $u \notin P$.

## 数学代写|交换代数代写commutative algebra代考|A source of examples: monomial ideals

One of the most important examples of a ring in this book is a polynomial ring in $n$ indeterminates, over a field $k$. Notation: $R=k\left[X_{1}, \ldots, X_{n}\right]$. This ring, along with its residue class rings will be thoroughly examined in forthcoming sections. Here, one wishes to single out a particular family of ideals in $R$, which has a distinctive role throughout modern commutative algebra and its computational side. This is the class of monomial ideals, to be briefly surveyed now.

An ideal $I \subset R=k\left[X_{1}, \ldots, X_{n}\right]$ is called a monomial ideal if it can be generated by a finite set of monomials $\mathbf{X}^{\mathbf{a}}=X_{1}^{a_{1}} \cdots X_{n}^{a_{n}}$, for varying $\mathbf{a}=\left(a_{1}, \ldots, a_{n}\right) \in \mathbb{N}^{n}$. The support of a such a monomial $\mathbf{X}^{\mathrm{a}}$ is the set of variables $X_{i}$ (or their respective indices) such that $a_{i}>0$. Denote by $\sqrt{\mathbf{X}^{\mathbf{a}}}$ the product of the variables in the support of the monomial $\mathbf{X}^{\mathbf{a}}$.

Let $I \subset R$ be an ideal. A basic criterion for $I$ to be a monomial ideal is that, whenever $f \in I$ then every nonzern term (= monomial afferted hy a coefficient from $k$ ) of $f$ also belongs to $I$. Moreover, given a set $\mathbf{u}$ of monomial generators of a monomial ideal $I$, if $f \in I$ then every nonzero term of $f$ is a multiple of some monomial in $\mathbf{u}$. This is besides a great facilitator in the calculations.

In particular, one advantage of a monomial ideal is that one needs not dancing around with different sets of minimal generators. Precisely, if $G$ and $H$ are sets of monomial generators of an ideal, which are both minimal with respect to divisibility (i.e., if $u, v \in G$, then neither $u \notin(v)$, nor $v \notin(u)$ ), then $G=H$.

Given two monomials $u, v, \operatorname{gcd}(u, v)$ denotes their greatest common divisor and $\operatorname{lcm}(u, v)$, their least common multiple.

The class of monomial ideals is closed under most common ideal operations, certainly under the ones for arbitrary ideals.

## 数学代写|交换代数代写commutative algebra代考|Prime and primary ideals

(1) 给定一个,b∈R这样一个b∈我， 然后一个∈我或者b∈我.
(2) 给定的理想Ĵ,Ĵ′⊂R这样ĴĴ′⊂我， 然后Ĵ⊂我或者Ĵ′⊂我.
(3) 残基类环R/我没有适当的零除数。

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