Antinous was a Bithynian Greek youth and a favourite or beloved of the Roman emperor Hadrian. He was deified after his death, being worshipped in both the Greek East and Latin West, sometimes as a god and sometimes as a hero. Little is known of Antinous's life, although it is known that he was born in Claudiopolis, in the Roman province of Bithynia, he was introduced to Hadrian in 123, before being taken to Italy for a higher education. He had become the favourite of Hadrian by 128, when he was taken on a tour of the Empire as part of Hadrian's personal retinue. Antinous accompanied Hadrian during his attendance of the annual Eleusinian Mysteries in Athens, was with him when he killed the Marousian lion in Libya. In October 130, as they were part of a flotilla going along the Nile, Antinous died amid mysterious circumstances. Various suggestions have been put forward for how he died, ranging from an accidental drowning to an intentional human sacrifice or suicide. Following his death, Hadrian deified Antinous and founded an organised cult devoted to his worship that spread throughout the Empire.

Hadrian founded the city of Antinoöpolis close to Antinous's place of death, which became a cultic centre for the worship of Osiris-Antinous. Hadrian founded games in commemoration of Antinous to take place in both Antinoöpolis and Athens, with Antinous becoming a symbol of Hadrian's dreams of pan-Hellenism; the worship of Antinous proved to be one of the most enduring and popular of cults of deified humans in the Roman empire, events continued to be founded in his honor long after Hadrian's death. Antinous became associated with homosexuality in Western culture, appearing in the work of Oscar Wilde and Fernando Pessoa; the Classicist Caroline Vout noted that most of the texts dealing with Antinous's biography only dealt with him and were post-Hadrianic in date, thus commenting that "reconstructing a detailed biography is impossible". The historian Thorsten Opper noted that "Hardly anything is known of Antinous's life, the fact that our sources get more detailed the they are does not inspire confidence."

Antinous's biographer Royston Lambert echoed this view, commenting that information on him was "tainted always by distance, sometimes by prejudice and by the alarming and bizarre ways in which the principal sources have been transmitted to us." It is known that Antinous was born to a Greek family in the city of Claudiopolis, located in the Roman province of Bithynia in what is now north-west Turkey. The year of Antinous's birth is not recorded, although it is estimated that it was between 110 and 112 AD. Early sources record that his birthday was in November, although the exact date is not known, Lambert asserted that it was on 27 November. Given the location of his birth and his physical appearance, it is that part of his ancestry was not Greek. There are various potential origins for the name "Antinous". Another possibility is that he was given the male equivalent of Antinoë, a woman, one of the founding figures of Mantineia, a city which had close relations with Bithynia. Although many historians from the Renaissance onward asserted that Antinous had been a slave, only one of around fifty early sources claims that, it remains unlikely, as it would have proved controversial to deify a former slave in Roman society.

There is no surviving reliable evidence attesting to Antinous's family background, although Lambert believed it most that his family would have been peasant farmers or small business owners, thereby being undistinguished yet not from the poorest sectors of society. Lambert considered it that Antinous would have had a basic education as a child, having been taught how to read and write; the Emperor Hadrian spent much time during his reign touring his Empire, arrived in Claudiopolis in June 123, when he first encountered Antinous. Given Hadrian's personality, Lambert thought it unlikely that they had become lovers at this point, instead suggesting it probable that Antinous had been selected to be sent to Italy, where he was schooled at the imperial paedagogium at the Caelian Hill. Hadrian meanwhile had continued to tour the Empire, only returning to Italy in September 125, when he settled into his villa at Tibur, it was at some point over the following three years that Antinous became his personal favourite, for by the time he left for Greece three years he brought Antinous with him in his personal retinue.

Lambert described Antinous as "the one person who seems to have connected most profoundly with Hadrian" throughout the latter's life. Hadrian's marriage to Sabina was unhappy, there is no reliable evidence that he expressed a sexual attraction for women, in contrast to much reliable early evidence that he was sexually attracted to boys and young men. For centuries, sexual relations between a man and a boy had been acceptable among Greece's leisured and citizen classes, with an older erastes undertaking a caring sexual relationship with an eromenos and taking a key role in his education. Hadrian took Antinous as a favoured servant when they were aged about 13 respectively; such a societal institution of pederasty was not indigenous to Roman culture, although bisexuality was the norm in the upper echelons of Roman society by the early 2nd century and was socially accepted. It is known that Hadrian believed Antinous to

In mathematics, it is possible to combine several rings into one large product ring. This is done by giving the Cartesian product of a family of rings coordinatewise addition and multiplication; the resulting ring is called a direct product of the original rings. An important example is the ring Z/nZ of integers modulo n. If n is written as a product of prime powers, n = p 1 n 1 p 2 n 2 ⋯ p k n k, where the pi are distinct primes Z/nZ is isomorphic to the product ring Z / p 1 n 1 Z × Z / p 2 n 2 Z × ⋯ × Z / p k n k Z; this follows from the Chinese remainder theorem. If R = Πi∈I Ri is a product of rings for every i in I we have a surjective ring homomorphism pi: R → Ri which projects the product on the ith coordinate; the product R, together with the projections pi, has the following universal property: if S is any ring and fi: S → Ri is a ring homomorphism for every i in I there exists one ring homomorphism f: S → R such that pi ∘ f = fi for every i in I. This shows; when I is finite, the underlying additive group of Πi∈I Ri coincides with the direct sum of the additive groups of the Ri.

In this case, some authors call R the "direct sum of the rings Ri" and write ⊕i∈I Ri, but this is incorrect from the point of view of category theory, since it is not a coproduct in the category of rings: for example, when two or more of the Ri are nonzero, the inclusion map Ri → R fails to map 1 to 1 and hence is not a ring homomorphism. Direct products are commutative and associative, meaning that it doesn't matter in which order one forms the direct product. If Ai is an ideal of Ri for each i in I A = Πi∈I Ai is an ideal of R. If I is finite the converse is true, i.e. every ideal of R is of this form. However, if I is infinite and the rings Ri are non-zero the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal, not a direct product of ideals of the Ri; the ideal A is a prime ideal in R if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. However, the converse is not true. For example, the direct sum of the Ri form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal, a fortiori prime.

An element x in R is a unit if and only if all of its components are units, i.e. if and only if pi is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of Ri. A product of two or more non-zero rings always has nonzero zero divisors: if x is an element of the product whose coordinates are all zero except pi, y is an element of the product with all coordinates zero except pj where i ≠ j xy = 0 in the product ring. Direct product Herstein, I. N. Noncommutative rings, Cambridge University Press, ISBN 978-0-88385-039-8 Lang, Algebra, Graduate Texts in Mathematics, 211, New York: Springer-Verlag, p. 91, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001

Sylvan Tale is an action-adventure game developed and published by Sega for the Game Gear. It was released in Japan in 1995, however there is an English patch for the game released by Aeon Genesis; the music for the game was composed by Saori Kobayashi, who contributed to the soundtracks of Sega's Panzer Dragoon Saga and Panzer Dragoon Orta. Sylvan Tale follows the typical action-adventure game formula: The player controls a character called Zetts who must solve puzzles, fight enemies, talk to non-player characters in order to acquire special powers and items that will allow him to unlock new areas of the game world and solve the puzzles within. While the player may revisit areas, the game progresses in an linear fashion, as each area can only be accessed if Zetts has acquired a specific item or ability from the previous area. Sylvan Tale at SMSPower.org Fan translation at romhacking.net Sylvan Tale at MobyGames