The Center of a Ring and Subrings

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In summary, Z(R) is defined as the set of elements in R that commute with all elements in R. To show that Z(R) is a subring of R, one must show that it is closed under addition and multiplication. This can be done by showing that for any elements x and y in Z(R), x+y and xy are also in Z(R). This can be proven by using the definition of Z(R) and the axioms of a ring.
  • #1
lostinmath08
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1. The problem statement

Let R be a ring. The center of R is defines as follows:

Z(R)= {x E R where xy = yx for all y E R}

Show that Z(R) is a subring of R



The Attempt at a Solution



I know that rings have to follow 4 axioms
a) its an abelian group under addition
b) Closure (ab E R)
c) Associativity ((ab)c =a(bc)
d) Distributivity a(b+c)=ab+ac and (b+c)a= ba+ca

Do the axioms apply to sub rings as well? and how would u go about solving it?
 
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  • #2
In general, a subring of R is a subset of R which is a ring with structure comparable to R. So you don't actually have to show all the axioms because multiplication being associative and distributive is inherited just by being a subset of R. Similarly, some of the additive group structure is inherited from R (associativity, commutativity of addition in particular). In short, you just have to show that whatever subset that you're claiming is a subring is closed under addition, multiplication, and taking of additive inverses (or more compactly, closed under subtraction and multiplication).
 
  • #3
Exactly right. The axioms have to apply to the subring as well. Start proving them one by one. E.g. if a and b are in Z(R), is a+b in Z(R)?
 
  • #4
i) (for all or any) x,y E R implies x+(-y) E R
ii) (for all or any) x,y E R implies xy E R ( R is closed under mulitplication)

so using the requirements of a subring...this is what i came up with:

x-y=y-x
-y-y=-x-x
-2y=-2x
y=x

and vice versa.

The above is just to satisfy the first axiom.
am i on the right track or completely off?
 
  • #5
Off. You know x+y is in R. R is already a ring. You just want to show for x and y in Z(R), x+y is in Z(R). To do that you have to show that for any w in R, (x+y)*w=w*(x+y). Remember you can use that x and y are in Z(R). So x*w=w*x and y*w=w*y.
 
  • #6
By criterion theorem, S is a subring of R iff x-y E R and xy E R for all x, y E S so you must show these two are true.

So let x,y E Z(R) and r E R,

To show x-y E Z(R), you have to look at r(x-y).

by definition, rx=xr and ry=yr, so

r(x-y)= rx-ry = xr-yr= (x-y)r so x-yE Z(R). (This proves that Z(R) is an abelian subgroup of R under addition).

Next, you need to show xy E Z(R) by looking at r(xy).

r(xy)= (rx)y=x(ry)= (xy)r. (Proving Z(R) is closed under multiplication)

Thus by criterion theorem, Z(R) is a subgroup of R.
 

1. What is the center of a ring?

The center of a ring is the set of elements that commute with every element in the ring. In other words, the center is the set of elements that can be multiplied in any order with any element in the ring and still produce the same result.

2. How is the center of a ring different from a subring?

The center of a ring is a subset of the ring, while a subring is a subset of the ring that is also a ring itself. The center consists of elements that commute with all elements in the ring, while a subring must contain the identity element and be closed under addition and multiplication.

3. Can a ring have more than one center?

No, a ring can only have one center. This is because the center must contain all elements that commute with every element in the ring, and any subset of the center would not satisfy this property.

4. How is the center of a ring related to its subrings?

The center of a ring is a subring itself, as it is a subset of the ring that is closed under addition and multiplication. Additionally, a subring's center is a subset of the ring's center, as all elements in the subring must also commute with elements in the ring.

5. Why is the concept of the center of a ring important in mathematics?

The center of a ring is important because it allows us to study the structure and properties of a ring. It also helps in finding subrings and understanding their relationship with the ring. The center is also useful in proving theorems and solving equations in ring theory.

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