Math 113, Spring 2021

Takehome Assigment 3

（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。

（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，

（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

（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.

{同态φ：R→A使得S的元素映射到A×}

⇕{同态φ：S-1R→A}。

1个

（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

（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）

（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 φ1(q) is a prime ideal of R.
2. (b)  Suppose φ is surjective, and m S is a maximal ideal. Show that φ1(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 S1R.
1. (a)  Considerthesubset{(a,b):aR,bS}⊆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 S1R = {ab : a R,b S} be the set of equivalence classes of the relation described above. Define addition and multiplication on S1R by the rules:

a1+a2 = a1b2+a2b1 b1 b2 b1b2

a1×a2 = a1a2.

b1 b2 b1b2
Show that these rules make S1R 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 S1R by the rule ι(r) = 1r . Show that ι is a ring homomorphism, that ι(1R)=1S1R andthatifsSR,theι(s)isaunitinS1R. Provealsothatιis injective if and only if S contains no zero divisors (or zero),

(d) Show that S1R 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 ̃ : S1R A such that φ ̃ ι = φ.

S1R 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 ̃ : S1R A}.

1

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] := S1R. 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. pR 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. mR 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 1ry R× for every y R. (Parts (b) and (c) might help!)

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