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An element a of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones (i.e., the units and the products ua, where u is a unit). Equivalently, an ...

A proper ideal of a ring that is not the intersection of two ideals which properly contain it. In a principal ideal domain, the ideal I=<a> is irreducible iff a=0 or a is an ...

A ring in which the zero ideal is an irreducible ideal. Every integral domain R is irreducible since if I and J are two nonzero ideals of R, and a in I, b in J are nonzero ...

A submodule N of a module M that is not the intersection of two submodules of M in which it is properly contained. In other words, for all submodules N_1 and N_2 of M, N=N_1 ...

An algebraic variety is called irreducible if it cannot be written as the union of nonempty algebraic varieties. For example, the set of solutions to xy=0 is reducible ...

Let A be a unital C^*-algebra, then an element u in A is called an isometry if u^*u=1.

A homotopy from one embedding of a manifold M in N to another such that at every time, it is an embedding. The notion of isotopy is category independent, so notions of ...

A line in the complex plane with slope +/-i. An isotropic line passes through either of the circular points at infinity. Isotropic lines are perpendicular to themselves.

A special ideal in a commutative ring R. The Jacobson radical is the intersection of the maximal ideals in R. It could be the zero ideal, as in the case of the integers.

In a lattice, any two elements a and b have a least upper bound. This least upper bound is often called the join of a and b, and is denoted by a v b. One can also speak of ...