# subring

Let $(R,+,*)$ a ring. A subring is a subset $S$ of $R$ with the operations^{} $+$ and $*$ of $R$ restricted to $S$ and such that $S$ is a ring by itself.

Notice that the restricted operations inherit the associative and distributive properties of $+$ and $*$, as well as commutativity of $+$.
So for $(S,+,*)$ to be a ring by itself, we need that $(S,+)$ be a subgroup^{} of $(R,+)$ and that $(S,*)$ be closed.
The subgroup condition is equivalent^{} to $S$ being non-empty and having the property that $x-y\in S$ for all $x,y\in S$.

A subring $S$ is called a left ideal^{} if for all $s\in S$ and all $r\in R$ we have $r*s\in S$. Right ideals are defined similarly, with $s*r$ instead of $r*s$.
If $S$ is both a left ideal and a right ideal, then it is called a two-sided ideal. If $R$ is commutative^{}, then all three definitions coincide. In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroups^{} in group theory.

Example:

Consider the ring $(\mathbb{Z},+,\cdot $). Then $(2\mathbb{Z},+,\cdot )$ is a subring, since the difference^{} and product^{} of two even numbers is again an even number.

Title | subring |

Canonical name | Subring |

Date of creation | 2013-03-22 12:30:19 |

Last modified on | 2013-03-22 12:30:19 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20-00 |

Classification | msc 16-00 |

Classification | msc 13-00 |

Related topic | Ideal |

Related topic | Ring |

Related topic | Group |

Related topic | Subgroup |

Defines | ideal |