What does it mean for a Ring to be Stabilized by a map

In summary, the conversation discusses a problem related to division rings, their centers, and division subrings. The problem asks to show that if a division subring is stabilized by every map in the division ring, it is either equal to the division ring or a subset of the center of the division ring. The terminology used in the problem, specifically regarding the stabilization of a subring by a map, may be unclear and could potentially be clarified by using more common terminology.
  • #1
AcidRainLiTE
90
2

Homework Statement


Let D be a division ring, C its center and let S be a division subring of D which is stabilized by every map x -> dxd-1, d≠0 in D. Show that either S = D or S is a subset of C.


2. The attempt at a solution
I haven't actually started working on it yet because I am not sure what it asking. What does it mean for a subring to be stabilized by a map? And what is x? Is it an element of D?

I am familiar with the stabilizer of an element, but the above terminology is confusing me.
 
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  • #2
It means that dSd-1=S for all nonzero d. (same definition as the stabilizer of an element but with the set S substituted in its place!)

I think a more common terminology would be to say

"...and let S be a division subring of D which is stable under the action by every map x -> dxd-1, d≠0 in D"
 

Related to What does it mean for a Ring to be Stabilized by a map

1. What is a stabilized ring?

A stabilized ring is a mathematical concept in which a ring is mapped to itself by a specific map or function. This means that the elements of the ring do not change when the map is applied, and the ring is said to be "fixed" or "stabilized" by the map.

2. What does it mean for a ring to be stabilized by a map?

For a ring to be stabilized by a map, it means that the map preserves the structure and elements of the ring. In other words, when the map is applied to the ring, the elements of the ring do not change and the ring remains unchanged.

3. How is a ring stabilized by a map?

A ring can be stabilized by a map through a process called ring homomorphism. This means that the map preserves the ring's operations of addition and multiplication, as well as the ring's identity and inverse elements. The map must also preserve the ring's structure and properties.

4. What is the importance of a stabilized ring?

A stabilized ring has many applications in mathematics and science. It allows for the study and manipulation of mathematical structures and can be used to prove theorems and solve equations. In addition, stabilized rings are used in fields such as abstract algebra and algebraic geometry.

5. Can any ring be stabilized by a map?

No, not all rings can be stabilized by a map. The map must preserve the structure and properties of the ring, which means that the ring must have certain properties such as being commutative or having a multiplicative identity. If a ring does not have these properties, it cannot be stabilized by a map.

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