1

Let {K,+,*} be commutative ring with unit. Define new operations ¤ and o so x¤y=x+y-1 and xoy=x+y-xy. Show that {K,¤,o} is a ring.

How can I show that the opeartions are closed for K? How can I show that x+y-1 belongs to K and x+y-xy belongs to K?

I manged to prove the other steps.

Erika
  • 387
  • How do you show the operations are closed? It is obvious that they are since the operations are defined exclusively using other operations which are known to be closed. – JMoravitz Jan 05 '21 at 18:18
  • I understand that x+y belongs to K, but why does x+y-1 belong to k for instance? – Erika Jan 05 '21 at 18:20
  • 2
    $1$ is the notation for the unit of the original ring. $-$ is the notation for the "additive" inverse, etc... – JMoravitz Jan 05 '21 at 18:21
  • 2
    $x+y-1$ is the same as $(x+y)-1$. You already noted that for $x,y\in K$, $x+y\in K$. $1$ is the unit and is in $K$, so you have an element in $K$ ($x+y$) and you're subtracting an element in $K$ ($1$) from it. What can you conclude? – Cameron Williams Jan 05 '21 at 18:22

1 Answers1

1

Use the fact that $+$ and $\cdot$ are already closed over $K.$

We begin by showing $x,y\in K$ implies $x+y-1\in K.$ Namely, if $x,y\in K,$ then we know $x+y\in K.$ Then $-1\in K$ as well, so $(x+y)+(-1)\in K$ as well.

Nowe we show $x,y\in K$ implies $x+y-xy\in K.$ Again, $x,y\in K$ implies $x+y\in K$ and $x\cdot y\in K,$ so $-x\cdot y\in K$ as well, so $(x+y)+(-xy)\in K.$

Derive Foiler
  • 709