Prime Ring - Properties

Properties

  • A commutative ring is a prime ring if and only if it is an integral domain.
  • A ring is prime if and only if its zero ideal is a prime ideal.
  • A non-trivial ring is prime if and only if the monoid of its ideals lacks zero divisors.
  • The ring of matrices over a prime ring is again a prime ring.

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Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)