In the context of quantum information theory, the following structure seems to be of interest--

"... the full two-by-two matrix ring with entries in GF(2), M₂(GF(2))-- the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

-- "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

This ring is another way of looking at the 16 elements of the affine space A₄(GF(2))
over the 2-element field.  (Arrange the four coordinates of each
element-- 1's and 0's-- into a square instead of a straight line, and
regard the resulting squares as matrices.)  (For more on A₄(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.) 
Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring.  This system of 35 objects has
a subconfiguration isomorphic to the (2,2) generalized quadrangle W₂ (which occurs naturally as a subconfiguration of PG(3,2)-- see Inscapes.)

Saniga concludes:

"We have demonstrated that the basic properties of a system
of two interacting spin-1/2 particles are uniquely embodied in the
(sub)geometry of a particular projective line, found to be equivalent
to the generalized quadrangle of order two. As such systems are the
simplest ones exhibiting phenomena like quantum entanglement and
quantum non-locality and play, therefore, a crucial role in numerous
applications like quantum cryptography, quantum coding, quantum
cloning/teleportation and/or quantum computing to mention the most
salient ones, our discovery thus

• not only offers a principally new geometrically-underlined insight into their intrinsic nature,
• but also gives their applications a wholly new perspective
• and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."

The previous entry dealt with Plato's myth of the ring of Gyges that
conferred invisibility. Another legendary ring, from Hermann
Hesse, with some background from Carl Jung:

From C. G. Jung, Collected Works (Princeton U. Press), Volume 12--
Psychology and Alchemy (1944)-- Part II-- "Individual Dream Symbolism in
Relation to Alchemy"-- Chapter 3, "The Symbolism of the Mandala"-- as
quoted in Jung, Dreams, published by Routledge, 2001-- Page 265--

"... the dreamer is wandering about in a dark cave, where a battle is
going on between good and evil. But there is also a prince who knows
everything. He gives the dreamer a ring set with a diamond....

Visual impression (waking dream):

The dreamer is falling into the abyss. At the bottom there is a bear
whose eyes gleam alternately in four colours: red, yellow, green, and
blue. Actually it has four eyes that change into four lights. The bear
disappears and the dreamer goes through a long dark tunnel. Light is
shimmering at the far end. A treasure is there, and on top of it the
ring with the diamond. It is said that this ring will lead him on a
long journey to the east."

Hermann Hesse, The Journey to the East (1932):

"'... Despair is the result of each earnest attempt to go through life
with virtue, justice, and understanding and to fulfil their
requirements. Children live on one side of despair, the awakened on the
other side. Defendant H. is no longer a child and is not yet fully
awakened. He is still in the midst of despair. He will overcome it and
thereby go through his second novitiate. We welcome him anew into the
League, the meaning of which he no longer claims to understand. We give
back to him his lost ring, which the servant Leo has kept for him.'

The Speaker then brought the ring, kissed me on the cheek and placed
the ring on my finger. Hardly had I looked at the ring, hardly had I
felt its metallic coolness on my fingers, when a thousand things
occurred to me, a thousand inconceivable acts of neglect. Above all, it
occurred to me that the ring had four stones at equal distances apart,
and that it was a rule of the League and part of the vow to turn the
ring slowly on the finger at least once a day, and at each of the four
stones to bring to mind one of the four basic precepts of the vow. I
had not only lost the ring and had not once missed it, but during all
those dreadful years I had also no longer repeated the four basic
precepts or thought of them. Immediately, I tried to say them again
inwardly. I had an idea what they were, they were still within me, they
belonged to me as does a name which one will remember in a moment but
at that particular moment cannot be recalled. No, it remained silent
within me, I could not repeat the rules, I had forgotten the wording. I
had forgotten the rules; for many years I had not repeated them, for
many years I had not observed them and held them sacred-- and yet I had
considered myself a loyal League brother.

The Speaker patted my arm kindly when he observed my dismay and deep shame."

Bulletin of the American Mathematical Society,

Volume 31, Number 1, July 1994, Pages 1-14

Selberg's Conjectures
and Artin L-Functions

(pdf)

M. Ram Murty

Introduction

In its comprehensive form, an identity between an automorphic
L-function and
a "motivic" L-function is called a reciprocity law. The celebrated
Artin reciprocity
law is perhaps the fundamental example. The conjecture of
Shimura-Taniyama that
every elliptic curve over Q is "modular" is certainly the most
intriguing reciprocity
conjecture of our time. The "Himalayan peaks" that hold the secrets of
these
nonabelian reciprocity laws challenge humanity, and, with the visionary
Langlands
program, we have mapped out before us one means of ascent to those
lofty peaks.
The recent work of Wiles suggests that an important case (the
semistable case)
of the Shimura-Taniyama conjecture is on the horizon and perhaps this
is another
means of ascent. In either case, a long journey is predicted.... At the
1989 Amalfi meeting, Selberg [S] announced a series of
conjectures which
looks like another approach to the summit. Alas, neither path seems the
easier
climb....

[S] A. Selberg, Old and new
      conjectures and results
      about a class of Dirichlet series,
      Collected Papers, Volume II,
      Springer-Verlag, 1991, pp. 47-63.

Zentralblatt MATH Database on the above Selberg paper:

"These are notes of lectures presented at the Amalfi Conference on Number Theory, 1989.... There are various stimulating conjectures
(which are related to several other conjectures like the Sato-Tate
conjecture, Langlands conjectures, Riemann conjecture...).... Concluding
remark of the author: 'A more complete account with proofs is under preparation and will in time appear elsewhere.'"

Related material: Previous entry.

 Symmetry axes
of the square:

(See Damnation Morning.)

From the cover of the
 Martin Cruz Smith novel
Stallion Gate:

"That old Jew
gave me this here."

-- Dialogue from the
Robert Stone novel
A Flag for Sunrise.

Related material:

A Mass for Lucero,

Log24, Sept. 13, 2006--

-- and this morning's online
New York Times obituaries:

The above image contains summary obituaries for Cardinal Lustiger, Archbishop of Paris, 1981-2005, and for Sal Mosca,
jazz pianist and teacher. In memory of the former, see all of the
remarks preceding the image above. In memory of the latter, the remarks
of a character in Martin Cruz Smith's Stallion Gate on jazz piano may have some relevance:

"I
hate arguments. I'm a coward. Arguments are full of words, and each
person is sure he's the only one who knows what the words mean. Each
word is a basket of eels, as far as I'm concerned. Everybody gets to
grab just one eel and that's his interpretation and he'll fight to the
death for it.... Which is why I love music. You hit a C and it's a C
and that's all it is. Like speaking clearly for the first time. Like
being intelligent. Like understanding. A Mozart or an Art Tatum sits at
the piano and picks out the undeniable truth."