In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal containing no other non-zero two-sided ideal of R (Isaacs 2009, p. 190).
In other words, minimal right ideals are minimal elements of the partially ordered set (poset) of non-zero right ideals of R ordered by inclusion. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, and so the zero ideal could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal.
theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero...
theorem use minimal prime ideals. A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing...
performs a sequence of simplifications to find a locally minimalideal triangulation. Once a suitable ideal triangulation is found, SnapPea can try to find a...
a minimal left ideal in Mat(2, C {\displaystyle \mathbb {C} } ). In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of...
Any primitive ideal is prime. As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. A ring is a prime...
Maximal right/left/two-sided ideals are the dual notion to that of minimalideals. If F is a field, then the only maximal ideal is {0}. In the ring Z of integers...
no minimal generating set. The cardinality of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by...
mathematics, the term minimal prime may refer to Minimal prime ideal, in commutative algebra Minimal prime (recreational mathematics), the minimal prime number...
subgroup and also an ideal of ⟨ a ⟩ {\displaystyle \langle a\rangle } . It is called the kernel of a and it is the minimalideal of the monogenic semigroup...
since this ring is a local ring with its maximal ideal generated by x and y, and unique minimalideal generated by xy. For a field k, the three-dimensional...
module theory). The minimal elements of { Q i ∣ i } {\displaystyle \{{\sqrt {Q_{i}}}\mid i\}} are the same as the minimal prime ideals containing I {\displaystyle...
(algebra) Ring ideal Principal idealIdeal quotient Maximal ideal, minimalideal Primitive ideal, prime ideal, semiprime ideal Radical of an ideal Jacobson...
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much...
a commutative monoid under the operation ∗ {\displaystyle *} . The minimalideal of this monoid is then isomorphic to the group of recurrent configurations...
an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a...
Ideal observer theory is the meta-ethical view which claims that ethical sentences express truth-apt propositions about the attitudes of a hypothetical...
finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important role...
catenary. In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's...
the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m...
Arbitrary semigroup E Set of idempotents in S G Group of units in S I Minimalideal of S V Regular elements of S X Arbitrary set a, b, c Arbitrary elements...
Holy minimalism, mystic minimalism, spiritual minimalism, or sacred minimalism are terms, sometimes pejorative, used to describe the musical works of...
ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are...
algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences...
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