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Hi brainly users,
Here is your today's question in the attachment :-
Please answer it fast
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Answer:
2,3,4,5
The characteristic of a commutative ring R with 1 is defined as follows.
The characteristic of a commutative ring R with 1 is defined as follows.Let us define the map ϕ:Z→R by sending n∈Z to
The characteristic of a commutative ring R with 1 is defined as follows.Let us define the map ϕ:Z→R by sending n∈Z toϕ(n)={1+⋯+1⏟n times if n>00 if n=0–(1+⋯+1⏟−n times) if n<0.
3,6,7,10
Step-by-step explanation:
Let cc be the characteristic of an integral domain RR.
Then by the first isomorphism theorem with the ring homomorphism ϕ:Z→Rϕ:Z→R as above, we have an injective homomorphism
Z/cZ=Z/ker(ϕ)→R.Z/cZ=Z/ker(ϕ)→R.
Since RR is an integral domain, Z/cZZ/cZ is also an integral domain.
This yields that cZcZ is a prime ideal of ZZ.
I'm not sure Actually