Does the Ring for Proposing and Wedding Ring Have to Be Different?

Traditionally, there are two rings: the engagement ring and the wedding band. The engagement ring is typically the fancy, expensive ring presented during the proposal. The wedding band is the ring put on the woman's finger during the wedding ceremony. I see nothing wrong with having just a single ring for the overall marriage. I would discuss it with your significant other or one of their friends/family members.

1. Describe a subring of the ring of functions from a monoid to a ring

Yes, that's right. It might be helpful to go one step further and note that we can identify $R$ inside $R[S]$ by the function $r(x)=r$ if $x=e$ and $r(x)=0$ otherwise.It's OK: the subring generated by a subset $X$ of a ring is by definition a subring. If you really want to know what it looks like, it looks like finite sums of finite products of elements of $X$ (including the empty product, which is the identity element.)This is correct, and $(f_uf_v)(x)=0$ for any $xin Ssetminus u,v$. If $u

eq v$ then indeed $f_uf_v$ is nonzero on two elements.No... there is not any reason to expect that. In general $f_x$ is going to be nonzero on finitely many elements of $S$. If you can pick $10^6$ distinct elements of $S$ and add them this way, the resulting function is $1$ on those $10^6$ places.What is unusual here is that you've defined $R[S]$ to be the entire set of functions from $S$ into $R$. $R[S]$ is often used to denote the semigroup ring, but for semigroup rings you need to restrict attention to the functions which are finitely nonzero. In fact, I believe you need to add this restriction into what you've written, because otherwise you have no guarantee the sum you are using is defined (what if it's an infinite sum?)This would bring you into the normal definition of a semigroup ring. Then you can alternatively view the functions as "formal linear combinations" using $S$ as a basis. The function simply tells you what the coefficients of the linear combination are, and because the function has finite support, it is only a finite linear combination.Work to show that it is the group ring $mathbb Z_n[S]$ where $mathbb Z_n$ is the subring of $R$ generated by $1$. In terms of the definition you've given, it would be the functions $Sto mathbb Z_n$ of finite support. It's a simple matter to show that these elements form a subring generated by the particular elements you picked out.

2. Weak ring which is not a ring

A slight modification of the authors' suggestion can produce non-commutative examples, namely to take subsets of a non-commutative ring that happen to be closed under addition and multiplication. For example a proper two-sided ideal of a non-commutative ring.For the sake of concreteness, consider the ring of two-by-two integer matrices, a non-commutative ring. From this ring take the subset $S$ of matrices whose entries are even. Clearly $S$ is closed under addition and multiplication, and the properties of left- and right-distributivity are inherited from the larger ring.It lacks a multiplicative identity of course, so it is not a proper ring.

3. I can't figure out how to change my headlight (Honda Civic EX 2005)?

Pop the hood Reach in behind the headlamp in question. Locate plastic ring Give plastic ring a turn--it will only turn one way Pull bulb and ring out Unplug old head light. Take new headlight out of package--do not touch the glass If you touch the glass clean it with rubbing alcohol. Plug new head light in. Put plastic ring and headlight back in. Turn ring until it locks in That's it.

4. NEW Tongue ring ! Can I Put A New Ring In ?

you may wait till you are executed breastfeeding to get a clean physique piercing. physique piercings carry the prospect of an infection and ailment transmission, and that is very undesirable on your toddler. in simple terms wait some greater months till your toddler is weaned, then you definately wo not could difficulty. in case your tongue is already pierced, in simple terms putting the ring back in is high quality

5. Engagement ring or promise ring ?

Yep. You are freaking out. But.... into the army you go, it would seem. Actually, it is far too soon to put your girlfriend into this position. Sorry to say this but... it is INCREDIBLY selfish. She is a junior in High School and you want her to promise not to date anyone else for years because you decided to join the Army? That is not loving. It is incredibly selfish. Let this young woman grow up with all the experiences that a girl should have! Do not cripple her with your needs. It is cruel. No promise ring. No engagement ring. She is a high school student. Kiss her good by, tell her you will miss her, that you will write occasionally, you hope she will answer, and get on with your life. When you come home on leave now and then, you can give her a call.