Answer to Question #106460 in Abstract Algebra for Garima Ahlawat

Question

(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.

Answer 1

1.False.

Let $P_1 \ and \ P_2$ are two ideals of a ring $R$ .

Then , $P_1\cap P_2=\{ x\in R:x\in P_1 \ and \ x\in P_2 \}$

and $P_1P_2=\{a_1b_1+a_2b_2+.....a_nb_n:a_i\in P_1,b_i\in P_2,$ $n$ is a positive integer $\}$ .

Consider the ring $R=\Z$

Let $P_1=2\Z$ ,$P_2=2\Z$

Then,$P_1\cap P_2=2\Z$ .

Claim:$P_1.P_2=4\Z$

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

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

$\implies \ 4\Z \subset P_1.P_2$

Let $x\in P_1.P_2.$

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

for some ,$a_i,b_i \in \Z$ .

$\implies\ x= 4(a_1b_1+a_2b_2+.....+a_rb_r)\in 4\Z$ .

$\implies\ P_1.P_2\subset 4\Z$ .

Hence,$4\Z=P_1P_2.$

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

2.False.

$K×K=\{ (a,b):a,b\in K \}$

Addition and multiplication are defined as follows.

$(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2).$

$(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$ .

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

Let $(a_1,b_1)\neq(0,0)\in K×K$ .

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

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

But $(1,0)^{-1}$ 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

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