Question #106460
(1) if P1 and P2 are prime ideals of a ring R , then P1P2 = P1 intersection P2.
(2) if k is a field , so is k × k
(3) Q[x]/<x^6 + 17 > is a field of characteristic 6.
True/false. Justify.
1
Expert's answer
2020-03-27T14:24:17-0400

1.False.

Let P1 and P2P_1 \ and \ P_2 are two ideals of a ring RR .

Then , P1P2={xR:xP1 and xP2}P_1\cap P_2=\{ x\in R:x\in P_1 \ and \ x\in P_2 \}

and P1P2={a1b1+a2b2+.....anbn:aiP1,biP2,P_1P_2=\{a_1b_1+a_2b_2+.....a_nb_n:a_i\in P_1,b_i\in P_2, nn is a positive integer }\} .

Consider the ring R=ZR=\Z

Let P1=2ZP_1=2\Z ,P2=2ZP_2=2\Z

Then,P1P2=2ZP_1\cap P_2=2\Z .

Claim:P1.P2=4ZP_1.P_2=4\Z

Let x4Z. Then x=4a for some aZ.x\in 4\Z. \ Then \ x=4a \ for \ some\ a\in \Z.

Therefore,x=2a+2aP1.P2x=2a+2a\in P_1.P_2

     4ZP1.P2\implies \ 4\Z \subset P_1.P_2

Let xP1.P2.x\in P_1.P_2.

Then x=2a_1.2b_1+2a_2.2b_2+.........+2a_r2.b_r \

for some ,ai,biZa_i,b_i \in \Z .

     x=4(a1b1+a2b2+.....+arbr)4Z\implies\ x= 4(a_1b_1+a_2b_2+.....+a_rb_r)\in 4\Z .

     P1.P24Z\implies\ P_1.P_2\subset 4\Z .

Hence,4Z=P1P2.4\Z=P_1P_2.

Therefore,P1P2P1P2.P_1P_2\neq P_1 \cap P_2.


2.False.

K×K={(a,b):a,bK}K×K=\{ (a,b):a,b\in K \}

Addition and multiplication are defined as follows.

(a1,b1)+(a2,b2)=(a1+a2,b1+b2).(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2).

(a1,b1)(a2,b2)=(a1a2,b1b2)(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2) .

Clearly, (1,1)(1,1) is the multiplicative identity.

Let (a1,b1)(0,0)K×K(a_1,b_1)\neq(0,0)\in K×K .

Then (a1,b1)1(a_1,b_1)^{-1} does not exist in general.

for example, (1,0)(0,0).(1,0)\neq(0,0).

But (1,0)1(1,0)^{-1} does not exist.



3.False.

Every field is integral domain by definition and we know that the characteristic of an integral domain is zero or a prime number.

Hence ,6 is not a characteristics of the given field

Q[x]<x6+17>\frac{Q[x]}{<x^6+17>}













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