Ring with division

Ring with division . In mathematics , especially abstract algebra , the definition of rings requires that there be a non-empty set and two binary operations called, generally, addition and multiplication; with respect to the first, the set is an abelian additive group, with respect to multiplication it is a requirement that it be a semi-group, it does not need to be commutative, nor does the element of multiplicative identity exist. But in the case of multiplicative identity and each non-null element has its multiplicative inverse, it constitutes a new and important type of ring.

ring with division [1] is a ring with unity, in which any non-null element has its multiplicative inverse.

Any algebraic body is a ring with commutative division. There are non-commutative division rings.

By Wedderburn’s theorem, every ring with finite division is a finite body.

Summary

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  • 1 Illustrative cases
  • 2 Features
  • 3 References and notes
  • 4 Sources

Illustrative cases

  • The ring of continuous real functions defined in [0; 1] with the sum f + g and the product fºg (x) = f [g (x)], it is not commutative, its multiplicative identity is I (x) = x.
  • 17= {0, 1,2, …, 15,16} (integer division residues by 17); where 2×9 = 1, 3×6 = 1, 4×13 = 1, 5×7 = 1, 8X15 = 1, 10×12 = 1, 11×14 = 1, 16×16 = 1, 1×1 = 1. Each nonzero element has its multiplicative inverse-
  • The ring of the real square matrices, M 2(R), with non-null determinant, is a ring with division, whose multiplicative identity is the 2×2 matrix, such that the elements that are not the main diagonal are 0 and in this, are 1.

characteristics

  • Every finite integer domain is a ring with division.
  • Finite fields (prime order) are rings with division
  • The center Z ‘of a division ring K is an algebraic body, and its elements are those that commute with any element of K.
  • Division rings can be classified according to their dimension on their center A: Z ‘

In the example of the ring of real quaterniums, the center is formed by all the elements of the form a (1,0,0,0) and an isomorphism can be established with the set R of all real numbers.

 

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