# 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.${\ displaystyle \ mathrm {ev} _ {0}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle f (0) = 0}$ 