Is Division Allowed in Rings and Fields?

The simplest example of a ring is that of the integers $mathbb Z$, and this is in some sense the motivating example of a ring.Division in $mathbb Z$ is highly restrictive - we can only divide $m$ by $n$ if $m$ is a multiple of $n$. As such, division is not defined as a binary operation - there is no function that gives a multiplicative inverse. The definition of a field is a ring such that every non-zero element has a multiplicative inverse. Be careful though - we still cannot define a multiplicative inverse on the whole field, since $0$ has no inverse!However, if $mathbb F$ is a field, then $mathbb F^*=mathbb Fsetminus 0$ is a group under multiplication, so does have a multiplicative inverse. As such, if $y$ is a non-zero element of a field, it has an inverse which is usually denoted $y^-1$.

1. engagement rings and wedding rings....HELP?

Two words - best friend. Relatives can not always help with this, but her best friend would know exactly what she wants...She may not have the same taste, but she will still KNOW what her friend likes

2. nipple rings breast feeding?

My friend had her nipple pierced and took it out to breastfeed because it was too painful. Some people have been able to keep them in, but I think a vast majority of people take them out. I guess it depends on the person and the pain they can tolerate? Some people start leaking earlier before the baby is born, and some do not start until after labor. It's really hard to say WHEN it will come as each person is different. Usually beforehand, but do not be alarmed if it does not . I would recommend grabbing some formula just in case.

3. 3 rings vs 4 rings pistons: which design is better?

There are trade offs, so no clear answer here.First of all, just to be anal and clear something up ... what you have is actually a 4-ring and a 5-ring piston. The oil control ring at the bottom actually consists of two rings and a spacer.The main trade off is, the 4-ring piston will have less drag on the cylinder walls, but the 5-ring piston will seal better. A better sealing piston will keep more power in the cylinder, but you will see additional frictional losses. Most engines I've seen with the 5-ring setup are for diesel engines. With the piston which is slightly domed, you will have a slightly higher compression ratio. You would have to do some measuring to figure out how much the difference would be. I do not know if you are assuming the pistons are virtually the same or if you have the specs on both. One of the things you need to check is the piston height difference. To do this, place a single wrist pin in the pin bore on both pistons. You will then be able to measure how much taller one piston is from the other. This will give you a better idea of how much one piston will stick up higher from the other. This may be important when installed, as the taller one may stick out of the bore above the deck surface or interfere with the head (I doubt it, but you should check ir). One other thing to think about is the weight of the piston. The lighter one may have a slight advantage over the heavier one as it would be able to allow the engine to rev a little higher without fear of something coming apart

4. Examples of rings with idempotent elements

Let $X$ be a disconnected space. Then the ring $C(X)$ of complex continuous functions has an idempotent.In a sense, this is a motivating example, because if $R$ has a nontrivial idempotent, then $Spec(R)$ is disconnected. Since elements of $R$ can be viewed (loosely) as functions, we can construct an element of $R$ which is "one" on one half of $Spec(R)$ and "zero" on the other. (To make this precise, one thinks in terms of the sheaf of regular functions on the scheme $Spec(R)$.)

5. Straight guys and cock rings?

I am a straight man and i have one on most of the time