1.False.
Let P1 and P2 are two ideals of a ring R .
Then , P1∩P2={x∈R:x∈P1 and x∈P2}
and P1P2={a1b1+a2b2+.....anbn:ai∈P1,bi∈P2, n is a positive integer } .
Consider the ring R=Z
Let P1=2Z ,P2=2Z
Then,P1∩P2=2Z .
Claim:P1.P2=4Z
Let x∈4Z. Then x=4a for some a∈Z.
Therefore,x=2a+2a∈P1.P2
⟹ 4Z⊂P1.P2
Let x∈P1.P2.
Then x=2a_1.2b_1+2a_2.2b_2+.........+2a_r2.b_r \
for some ,ai,bi∈Z .
⟹ x=4(a1b1+a2b2+.....+arbr)∈4Z .
⟹ P1.P2⊂4Z .
Hence,4Z=P1P2.
Therefore,P1P2=P1∩P2.
2.False.
K×K={(a,b):a,b∈K}
Addition and multiplication are defined as follows.
(a1,b1)+(a2,b2)=(a1+a2,b1+b2).
(a1,b1)(a2,b2)=(a1a2,b1b2) .
Clearly, (1,1) is the multiplicative identity.
Let (a1,b1)=(0,0)∈K×K .
Then (a1,b1)−1 does not exist in general.
for example, (1,0)=(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
<x6+17>Q[x]
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