数学代写 | Math 113 Takehome Assigment 3

Takehome Assigment 3

在这项作业中，我们建立了一些关于交换中素理想和最大理想的基本事实

整体环。在该分配中，所有环将是具有身份的交换环。

设φ：R→S为φ（1R）= 1S的交换单环之间的同态。

（a）让q是最理想的。证明φ-1（q）是R的素理想。

（b）假设φ是射影，且m⊆S是最大理想值。证明φ−1（m）为最大值

S的理想

（c）如果φ不具排斥性，请给出（b）部分的反例。

在课堂上，我们定义了一个环的良好乘法子集的分数环，即R的子集，该子集不包含零除数，并且在乘法下是封闭的。如果R是一元环，则可以更一般地定义这一点。如果将子集S⊆R在乘法下关闭并且包含1，则将其定义为乘法子集。在本练习中，我们将描述分数S-1R的环。

（a）考虑子集{（a，b）：a∈R，b∈S}⊆R×R。证明：
（a1，b1）〜（a2，b2）如果存在t∈S使得t（a1b2- b1a2）= 0，

是R上的等价关系。（a，b）的等价类将表示为ab。解释为什么S不包含零除数，那么这与类中定义的等价关系相同。

（b）令S-1R = {ab：a∈R，b∈S}是上述关系的等价类的集合。通过以下规则在S-1R上定义加法和乘法：

a1 + a2 = a1b2 + a2b1 b1 b2 b1b2

a1×a2 = a1a2。

b1 b2 b1b2
证明这些规则使S-1R成为具有身份的交换环。 （你必须

首先表明它们定义明确。然后证明圆环公理是满意的）

（c）通过规则η（r）= 1r来定义η：R→S-1R。证明ι是一个环同构，ι（1R）= 1S-1R和ifs∈S⊆R，the（s）isaunitinS-1R。证明当且仅当S不包含零除数（或零）时，它才是内射性的，

（d）证明S-1R满足以下通用性质。对于任何可交换的单位环A，以及环同态φ：R→A，使得每个s∈S都有φ（s）∈A×，则有一个唯一的态素φ̃：S-1R→A，使得φ̃◦= φ。

S-1Rι

φ
[RφA.

推论出存在双射：
{同态φ：R→A使得S的元素映射到A×}

⇕{同态φ：S-1R→A}。

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加州大学伯克利分校数学113，2021年春季

（e）令r∈R为非零，并考虑相乘集S = {1，r，r2，r3，···}。定义R [1 / r]：＝ S-1R。当且仅当r为幂幂时，证明R [1 / r] = 0。

3.在本练习中，我们计算交换单环R中所有素理想的交点。

（a）证明元素0包含在R的每个理想中。

（b）让r成为R的幂等元素。证明r包含在R的每个素理想中。

（c）相反，假设r不是幂等的。表明存在一些不包含r的理想理想。推论：

N（R）=􏱭p。 p⊆R素数

（提示：要找到这样的理想理想，请尝试将1（a）和2（e）应用于图ι：R→R [1 / r]。）

（d）推论积分域中所有素理想的交集为0理想。

（e）假设r在R的所有素理想中的交点。证明每个y∈R都有1 − ry∈R×（我们将在下面看到相反的情况通常是不正确的，但是我们可以表征满足此属性的所有元素）。

4.在本练习中，我们计算交换单环R中所有最大理想的交点。给定环R，我们将R的Jacobson根定义为：

J（R）＝􏱭m。最大m⊆R

（a）证明N（R）⊆J（R）。

（b）证明当且仅当元素r∈R不包含在任何最大值中时，它才是一个单位

理想的。

（c）假设m为最大理想且r∈R \ m。计算由m生成的理想值（m，r）

和河。

（d）证明3（e）中的条件实际上表征了Jacobson自由基中的元素！也就是说，证明且仅当每个y∈R的1−ry∈R×时r∈J（R）（（b）和（c）部分可能有帮助！）

Due Monday, April 26
In this assignment we establish some basic facts about prime and maximal ideals in commutative

unital rings. In this assignment all rings will be commutative rings with identity.

1. Let φ : R → S be a homomorphism between commutative unital rings with φ(1R) = 1S.
   1. (a)  Let q ⊆ S be a prime ideal. Show that φ−1(q) is a prime ideal of R.
   2. (b)  Suppose φ is surjective, and m ⊆ S is a maximal ideal. Show that φ−1(m) is a maximal

      ideal of S.

   3. (c)  Give a counterexample to part (b) if φ is not surjective.
2. In class we defined the ring of fractions for a good multiplicative subset of a ring, i.e., a subset of R which contains no zero divisors and is closed under multiplication. If R is a unital ring, then one can define this slightly more generally. We define a subset S ⊆ R to a be multiplicative subset if it is closed under multiplication and contains 1. In this exercise we will describe the ring of fractions S−1R.
   1. (a)  Considerthesubset{(a,b):a∈R,b∈S}⊆R×R. Provethat:
      (a1, b1) ∼ (a2, b2) if there exits t ∈ S such that t(a1b2 − b1a2) = 0,

      is an equivalence relation on R. The equivalence class of (a, b) will be denoted ab . Explain why if S contains no zero divisors, this is the same equivalence relation as the one defined in class.

   2. (b)  Let S−1R = {ab : a ∈ R,b ∈ S} be the set of equivalence classes of the relation described above. Define addition and multiplication on S−1R by the rules:

      a1+a2 = a1b2+a2b1 b1 b2 b1b2

      a1×a2 = a1a2.

b1 b2 b1b2
Show that these rules make S−1R into a commutative ring with identity. (You must

first show that they are well defined. Then show that the ring axioms are satisfied)

(c) Define ι : R → S−1R by the rule ι(r) = 1r . Show that ι is a ring homomorphism, that ι(1R)=1S−1R andthatifs∈S⊆R,theι(s)isaunitinS−1R. Provealsothatιis injective if and only if S contains no zero divisors (or zero),

(d) Show that S−1R satisfies the following universal property. For any commutative unital ring A, and ring homomorphisms φ : R → A such that φ(s) ∈ A× for every s ∈ S, there is a unique morphism φ ̃ : S−1R → A such that φ ̃ ◦ ι = φ.

S−1R ι

φ ̃
R φ A.

Deduce that there is a bijection:
{Homomorphisms φ : R → A such that elements of S map to A×}

⇕ {Homomorphisms φ ̃ : S−1R → A}.

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University of California, Berkeley Math 113, Spring 2021

(e) Let r ∈ R be nonzero and consider the multiplicative set S = {1, r, r2, r3, · · · }. Define R[1/r] := S−1R. Show that R[1/r] = 0 if and only if r is nilpotent.

3. In this exercise we calculate the intersection of all the prime ideals in a commutative unital ring R.

1. (a)  Show that the element 0 is contained in every ideal of R.
2. (b)  Let r be a nilpotent element of R. Show that r is contained in every prime ideal of R.
3. (c)  Conversely, suppose r is not nilpotent. Show that there is some prime ideal not contain- ing r. Deduce that:

   N(R) = 􏱭 p. p⊆R prime

   (Hint: To find such a prime ideal, try applying 1(a) and 2(e) to the map ι : R → R[1/r].)

4. (d)  Deduce that the intersection of all the prime ideals in an integral domain is the 0 ideal.
5. (e)  Suppose that r is in the intersection of all the prime ideals of R. Show that 1 − ry ∈ R× for every y ∈ R. (We will see below that the converse is not true in general, but that we can characterize all elements satisfying this property).

4. In this exercise we calculate the intersection of all the maximal ideals in a commutative unital ring R. Given a ring R, we define the Jacobson radical of R to be:

J(R) = 􏱭 m. m⊆R maximal

1. (a)  Show that N(R) ⊆ J(R).
2. (b)  Show that an element r ∈ R is a unit if and only if it is not contained in any maximal

   ideal.

3. (c)  Suppose m is a maximal ideal and r ∈ R \ m. Compute the ideal (m, r) generated by m

   and r.

4. (d)  Prove that the condition from 3(e) actually characterizes elements in the Jacobson Rad- ical! That is, prove that r ∈ J(R) if and only if 1−ry ∈ R× for every y ∈ R. (Parts (b) and (c) might help!)