I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:
"Let be a semiprime ideal of a semigroup , and let . Letting , we obtain an m-system disjoint from ."
For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup with the property that for any . An m-system is a non-empty subset of S such that there exists such that .
What I'm trying to understand is why should be disjoint from . I understand why it is an m-system, but I don't understand why, say, can't be in . Any insights would be appreciated.
I'm using Introduction to Semigroups by Mario Petrich. It' pretty old (1973), so it's not entirely unlikely that the terminology is out of date.
The definition you gave is given in my book as the definition of a completely semiprime ideal. It makes a distinction between semiprime and completely semiprime. I agree, it's pretty trivial why M and I would be disjoint in that case, but my book only says that I is semiprime.
Are -systems defined on general semigroups, or on Monoids? The great John Howie's book only seems to define them for monoids (unless I am missing something). In monoids everything here works - the condition given by Tonio follows from spoon737's definition, and so the result holds...
Also, can you clarify your definition of an -system? I mean, do you mean to say instead of, say, ?
My book's definition of an m-system is exactly as stated: given a non-empty subset of a semigroup , is an m-system if for any there exists such that . So yes, is required to be in . I have yet to come across a definition for -system.
I think I've figured it out, though. After searching around a bit online, I found that my book's definition of a semiprime ideal is equivalent to the following:
An ideal is semiprime if for any ideal , implies .
Technically, the result I found at PlanetMath was given for rings, but it was easy enough to see that this was the same for semigroups. Thinking of it this way made the result much more obvious. However, since the author makes no hint of this equivalent definition, yet he still states that and are disjoint in such a way that seems too trivial to require proof, I have to wonder if it really does follow immediately from his definition and I'm just missing something.
this is for spoon737, the OP:
if anything in my answer is unclear for you, you should ask me in this website not asking it somewhere else! that's just not right! you (almost) copied my answer in there and, even worse,
asked them to confirm it!! that's a big insult to me! haha ... ok, i forgive you. i just wanted to tell you that what you did was wrong.