Maximum ideal

Maximum ideal is a term from algebra .

definition

Let it be a ring . Then an ideal is called maximal if there is a maximal element in the set of all real ideals semi-ordered by the (set-theoretical) inclusion . That means for every real ideal : ${\displaystyle R}$ ${\displaystyle {\mathfrak {m}}\subsetneq R}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle \subseteq }$ ${\displaystyle {\mathfrak {a}}\subsetneq R}$

From follows ${\displaystyle {\mathfrak {a}}\supseteq {\mathfrak {m}}}$ ${\displaystyle {\mathfrak {a}}={\mathfrak {m}}.}$

In other words:

A true ideal is called maximal when there is no other true ideal of that contains entirely. ${\displaystyle {\mathfrak {m}}\subsetneq R}$ ${\displaystyle R}$ ${\displaystyle {\mathfrak {m}}}$

Remarks

• The same applies to left and right ideals.
• With the help of Zorn's lemma one can show that every true ideal in a ring with one element 1 is contained in a maximal ideal.
• From this it follows in turn that every element of a commutative ring with 1 that is not a unit must be contained in a maximal ideal. In non-commutative rings this is i. A. wrong, as the example of the die rings over (oblique) bodies shows.
• Let be an ideal of the commutative ring with 1. The factor ring is a field if and only if is maximal. In particular, this means: The image of a ring homomorphism is a body if and only if its core is maximal. ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle R}$ ${\displaystyle R/{\mathfrak {m}}}$ ${\displaystyle {\mathfrak {m}}}$
• Rings can contain several maximal ideals. A ring that has only a single maximum left or right ideal is called a local ring . This is a two-sided ideal , and the quotient ring is called the residue field of the ring called. ${\displaystyle R/{\mathfrak {m}}}$ ${\displaystyle R\,}$
• A maximal (two-sided) ideal of a ring is prime if and only if . In particular, is prime if contains a one element. ${\displaystyle {\mathfrak {m}}\subseteq R}$ ${\displaystyle R}$ ${\displaystyle RR\nsubseteq {\mathfrak {m}}}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle R}$

Examples

• In the ring of whole numbers , every prime ideal except the zero ideal is maximal. However, this is generally not true; Integrity rings with this property are called (if they are not bodies) one-dimensional . All main ideal rings have this property. ${\displaystyle \mathbb {Z} }$
• Let be the ring of continuous functions on the real numbers with pointwise multiplication. Consider the ring homomorphism ${\displaystyle C(\mathbb {R} )}$

${\displaystyle \mathrm {ev} _{0}\colon C(\mathbb {R} )\rightarrow \mathbb {R} ,\quad f\mapsto f(0).}$

In other words: the mapping that evaluates every function at position 0. The image of is , therefore, a body. Thus the core, i.e. the set of all functions with , is a maximum ideal. ${\displaystyle \mathrm {ev} _{0}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f(0)=0}$