If the compromise was a rebuke to the most radical vision of the liberty of the Church, on at least one point its implication was firm and unmistakable: The king, even an emperor, was a layman.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If Anglo-Saxon England was already changing before the invasion, with the introduction of feudalism or castles or the changes in society, then the conquest was important but not a radical change.

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical.

If the rate of increase of life span can be raised with these technologies to a level of twelve months increase per year, this is defined as effective biological immortality and is the goal of radical life extension.

If usage preferences are conservative, prescription might ( appear to ) be resistant to language change ; if the usage preferences are radical, prescription may produce neologisms.

If an ideal I coincides with its own radical, then I is called a radical ideal or semiprime ideal.

If " Lin Biao and the Gang of Four ; that is, the radical leftists " had been able to assure the people food and clothing, he said, they would not have been so easy to overthrow.

If surgical therapy ( i. e., radical prostatectomy ) is successful at removing all prostate tissue ( and prostate cancer ), PSA becomes undetectable within a few weeks.

If Wieland's productions were radical, Wolfgang's were regressive.

If R is commutative, the Jacobson radical always contains the nilradical.

If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I.

If the ring is noetherian, then the Levitzki radical is itself a nilpotent ideal, and so is the unique largest left, right, or two-sided nilpotent ideal.

If the goal of 12x would be met, more radical changes would be needed, and a " high tech " approach would have to be used.

If path B occurs then no SKIE should be observed as the free radical intermediate does not change hybridization.

If the reaction is not terminated, but instead the radical group ( s ) go on to react further, the steps where new radicals are formed and then react is collectively known as propagation ( 4, 5 ) because a new radical is created available for secondary reactions.

If G is reductive then its radical is a torus.

If W is a subset of V, then the orthogonal complement is the set of all vectors orthogonal with every vector in W: it is a subspace of V. When B is non-degenerate, so that the radical of B is trivial, the dimension of

Let J ( R ) be the Jacobson radical of R. If U is a right module over a ring, R, and I is an right ideal in R, then define U · I to be the set of all ( finite ) sums of elements of the form u · i, where · is simply the action of R on U. Necessarily, U · I is a submodule of U.

If neighboring districts believe a particular change is too radical, the offended district could break off communications and refuse to recognize them as fellow Amish.

If working in a zinc mine, which he once did for 87-1/2 cents an hour, paid more than playing center field for the Yankees, Mantle would work in a zinc mine.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.

If one cube has an A atom at its center, the atom is on a site usually occupied by a B atom, and is thus an antisite defect.

If the panegyric detailing the ceremony implied that the true center of the Empire was not Rome, but where the Emperor sat ("... the capital of the Empire appeared to be there, where the two emperors met "), it simply echoed what had already been stated by the historian Herodian in the early third century: " Rome is where the emperor is ".

If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity.

If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse.

Formulae connecting a tangential angle, the angle anchored at the ellipse's center ( called also the polar angle from the ellipse center ), and the parametric angle t < ref > If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle is the geocentric latitude, and parametric angle t is a parametric ( or reduced ) latitude of auxiliary circle </ ref > are:

If the applied voltage is large enough, the liquid crystal molecules in the center of the layer are almost completely untwisted and the polarization of the incident light is not rotated as it passes through the liquid crystal layer.

If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade, g, symmetry.

If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade, u, symmetry.

If all the particles are moving, the center of mass will generally be moving as well.

If the center of mass is moving at velocity, the momentum is:

If the top has mass and its center of mass is at a distance from the pivot point, its gravitational potential is

If G is a p-group, then so is G / Z, and so it too has a nontrivial center.

If the cross stands at the center of Paul's theology, so does the Resurrection: unless the one died the death of all, the all would have little to celebrate in the resurrection of the one.

If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.

If the planes don't pass through the sphere's center, then the intersection is called spheric section.