Question

What is commutative division ring?

Answer 1

Answer:

The center of a division ring is commutative and therefore a field. Every division ring is therefore a division algebra over its center. ... Every field is, of course, one-dimensional over its center. The ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

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Answer 2

Answer:  A division ring is a ring where each nonzero element has a multiplicative inverse and where 0 ≠ 1 . A skew field is a noncommutative division ring and a field is a commutative division ring.

Explanation:

• A commutative ring in mathematics is one in which multiplication is commutative. Commutative algebra refers to the study of commutative rings.
• The study of ring properties that are not unique to commutative rings is known as noncommutative algebra. A noncommutative ring is also a division ring.
• If and only if it is a field, it is commutative. For instance, according to Wedderburn's little theorem, all finite division rings are commutative, and as a result, all finite fields are finite fields.
• If a commutative ring with identity has no zero divisors, it is said to have an integral domain. We say that an element an is a unit if it has a multiplicative inverse in a ring R with identity. 
• If a ring R contains only nonzero elements, then R is referred to be a division ring.

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