True or false:
(1) Every integral domain is a field
(2) every field is an integral domain
(3) the ring $\mathbb Z$ is a field.
(4) the ring $\mathbb Z/(17)$ is a field.
(5)The set $\{[0], [2], [4]\}$ is a subring of $\mathbb Z/(6)$
So I am having a little trouble with the whole concept of rings. I know the definition of field, integral domain, ring ect., but I am just having trouble applying it. I made up these questions based off the lecture notes just to maybe help me out. If anyone can shed a little insight I would appreciate it. My thoughts: 1-false 2-true 3-false 4-true 5-false Again not sure on the answers of interpreting them. Just going by what I understood in lecture.
(1) False, consider $\mathbb{Z}$.
(2)True, a field is an integral domain in which every element is a unit.
(3) False, check the elements that have inverses (HINT: There are only two elements that are units in $\mathbb{Z}$)
(4)True, you know that it is an integral domain, now check that every element is a unit.
(5)See if you can work this one out on your own.
Hope this helps!