Polynomial identity ring

In mathematics, in the subfield of ring theory, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z<X₁, X₂, ..., X_N>, over the ring of integers in N variables X₁, X₂, ..., X_N such that for all N-tuples r₁, r₂, ..., r_N taken from R it happens that

Strictly the X_i here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra.

If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1.

Every commutative ring is a PI-ring, satisfying the polynomial identity XY - YX = 0. Therefore PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.

*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

for all x, y in A.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.

Ring (Suzuki novel)

Ring (リング, Ringu) is a Japanese mystery horror novel by Koji Suzuki, first published in 1991, and set in modern-day Japan. It was the basis for a 1995 television film (Ring: Kanzenban),a television series (Ring: The Final Chapter), a film of the same name (1998's Ring), and two remakes of the 1998 film: a South Korean version (The Ring Virus) and an American version (The Ring).

Plot synopsis

After four teenagers mysteriously die simultaneously in Tokyo, Kazuyuki Asakawa, a reporter and uncle to one of the deceased, decides to launch his own personal investigation. His search leads him to "Hakone Pacific Land", a holiday resort where the youths were last seen together exactly one week before their deaths. Once there he happens upon a mysterious unmarked videotape. Watching the tape, he witnesses a strange sequence of both abstract and realistic footage, including an image of an injured man, that ends with a warning revealing the viewer has a week to live. Giving a single means of avoiding death, the tape's explanation ends suddenly having been overwritten by an advertisement. The tape has a horrible mental effect on Asakawa, and he doesn't doubt for a second that its warning is true.

Ring (film)

Ring (リング, Ringu) is a 1998 Japanese psychological horror film directed by Hideo Nakata, adapted from the novel Ring by Kôji Suzuki, which in turn draws on the Japanese folk tale Banchō Sarayashiki. The film stars Nanako Matsushima, Hiroyuki Sanada, and Rikiya Ōtaka. The film follows TV reporter and single mother Reiko who is caught up in a series of deaths surrounding a cursed video tape.

Production took approximately 9 months.Ring and its sequel Rasen were released in Japan at the same time.

After release, Ring inspired numerous follow-ups within the Ring franchise and triggered a trend of Western remakes.

Plot

Two teenagers, Masami (Hitomi Satō) and Tomoko (Yūko Takeuchi), talk about a videotape recorded by a boy in Izu which is fabled to bear a curse that kills the viewer seven days after watching. Tomoko reveals that a week ago, she and three of her friends watched a weird tape and received a call after watching. Tomoko is killed by an unseen force as Masami watches, horrified.