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In my previous sessional exams, I was asked to prove these two:

1) Find a ring which doesn't have a maximal Ideal.

2) If a ring has unity, then it has a maximal Ideal.

About the first one we can think about the trivial ring where there are only two additive subgroups (one the group itself and the other containing the identity element only....). Is there some other case for the first one...

About the second one I don't know how to show it...

Kindly help...

user26857
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patang
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1 Answers1

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1) The simplest example is the zero ring. It has no maximal ideal (recall that maximal ideals are required to be proper ideals). For a more interesting example, consider the ring $(\mathbb{Q},+,0)$, where $+$ is the usual addition and $0$ is the zero multiplication. It is known that $(\mathbb{Q},+)$ has no maximal subgroups, which implies that $(\mathbb{Q},+,0)$ has no maximal ideals.

2) is not correct. As I've said, the zero ring (and this is unital with $0=1$) has no maximal ideal. But if $R$ is a non-zero unital ring, then $R$ has a maximal ideal. In fact, one may apply Zorn's Lemma to the partial order of proper ideals of $R$. The crucial observation is that for any chain $\mathcal{K}$ of proper ideals, their union $\bigcup \mathcal{K}$ is again a proper ideal. And this uses the existence of a unit and that an ideal is proper iff it does not contain the unit.

Martin Brandenburg
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  • $(\mathbb{Q},+)$ has no maximal subgroups ,if I'm asked to prove it in my exam,how shall I prove it.... – patang Oct 19 '14 at 11:18
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    You shall think about it at least a couple of minutes ... If you do not want to do this, check http://math.stackexchange.com/questions/234995/ – Martin Brandenburg Oct 19 '14 at 11:20
  • by zero multiplication you possibly mean multiplying any two elements in the ring gives zero... – patang Oct 19 '14 at 11:25
  • Yes. $\phantom{...}$ – Martin Brandenburg Oct 19 '14 at 11:26
  • I accpted the answer because 1.) is perfect...about 2.) I don't know about what is zorn's lemma so,can't understnd it.. – patang Oct 19 '14 at 11:29
  • http://en.wikipedia.org/wiki/Zorn%27s_lemma – Martin Brandenburg Oct 19 '14 at 12:18
  • Well, I have seen a definition of a unity in a ring as a non-zero element, which means that the zero ring is then a ring without unity. I know this is strange, but it is exactly for the reason that zero ring is too weird to be a unit ring. – yo' Oct 19 '14 at 16:51
  • The zero ring is a unital ring, and this is not weird, it is absolutely natural. If you want unital rings to be $\neq 0$, you will have to make case distinctions in many constructions of rings, which is absurd. I know that, unfortunately, there are still books and lectures in which $1 \neq 0$ is a general assumption. In order to understand some concept, one really also has to understand the so-called pathological cases. See also http://mathoverflow.net/questions/45951/sexy-vacuity – Martin Brandenburg Oct 19 '14 at 17:54