Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:

Then, for example, (−∞, 1 ) and ( 0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection ( 0, 1 ) would be as well.

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Pick a fixed subset L of U. Then the maps F and G, where F ( M ) is the intersection of L and M, and G ( N ) is the union of N and ( U

Then along North Avenue, Northside Drive, out Donald L. Hollowell Parkway, then south along the BeltLine to the intersection of Ralph David Abernathy and Cascade Ave.

Then the collection has the finite intersection property ( fip ), if any finite subcollection has non-empty intersection

Then A, B and their intersection A ∩ B are homotopy equivalent to circles, so the nontrivial part of the sequence yields

Then the intersection of A and B is an annulus, which is known to be homotopy equivalent to ( and so has the same fundamental group as ) a circle.

Then the intersection number of two closed curves on X has a simple definition in terms of an integral.

Then the intersection number of two closed curves, a and b, on X is defined as

Then the intersection number at the point x is

Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P ( W ) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space ( the subspace

Then SR 9 has an intersection with State Road 26, just west of Fairmount.

Then SR 9 enters Marion, where SR 9 has an intersection with U. S. Route 35 / State Road 22 followed by an intersection with State Road 37.

Then SR 9 leaves the bypass on the north side of Huntington and heads north towards Columbia City, passing through an intersection with State Road 114 and State Road 14.

Then just south of the Michigan state line, SR 9 has an intersection with the Indiana Toll Road ( Interstate 80 / Interstate 90 ).

Then elimination theory can be used to calculate the intersection of I with ( g < sub > 1 </ sub >) and ( g < sub > 2 </ sub >):

Then drop the line at the point of intersection with the curve on x-axis.

Then East and Southeasterly via Rita to Richloam thence Northeasterly via Riverland-Tarrytown and Mable to a junction with SR 33 in Mascotte and along SR 33 to intersection with SR 459 at Groveland.

Then the intersection

Starting from the lower south-west corner, at the intersection of Glenmore St and Collins Tce, the boundary goes up through Tinakori Hill, across through Weld St and along to Baker St. Then the boundary follows the west side of Thorndon Quay down until Hill St, where it goes across to Bowen St, Tinakori Road and Glenmore St ( until Collins Tce ).

Then Little Billy began shouting orders to round up the ponies and fill the water buckets and for the cooks to hurry up with the meal.

Then she turned the station wagon around and headed it back down the hill, with the village as her ostensible destination.

Then, with a shrug of pretended indifference, she took a compact from her purse and went through the motions of fixing her make-up.

Then, as he doubled, gasping, vomiting the breakfast he had so lately eaten, Lord straightened him with an uppercut.

Then, with a glory that almost wiped out the deep, downward sags in her careworn face, Matilda leaned over the wheel and shouted to Hez, who was stumbling along in the heat and the dust on the opposite side of the wagon `` Pa!!

Then, and only then, with the Jacksons and Dan as their true guests of honor, did the Harrows take time to catch up on the news.

Then he took off his wet boots and dropped down into the water to talk with the beasts, needing their comfort more than they needed his.

Then the Bonaventure seemed to disintegrate with a roar of live steam, geysers of sparks and flames, and a dense cloud of black-and-orange smoke.

Then she saw Ramey and her face was misshapen with bewilderment.

Then they were tumbling again, and the big man reached into the same pocket he had gone for earlier, and came up with a vicious switchblade.

Then suddenly we found ourselves in the middle of another fight, an irrational, an indecent, an undeclared and immoral war with our strongest ( and some had thought noblest ) ally.

Then, with staring eyes and lips drawn thin, Miriam said to the young woman, `` You are ugly -- uglier than you used to be, and you were always very ugly.

Then, after overtures to accept a settlement and go through with a divorce, Miriam gave a ghastly echo of Mrs. Micawber by suddenly stating, `` I will never leave Mr. Wright ''.

Then Miriam varied the senseless psychological warfare by suddenly withdrawing a suit for separate maintenance that had been pending, and asking for divorce on the grounds of cruelty, with the understanding that Wright would not contest it.

Then, in some way, this lack of faith in the cavalry became mixed up in his mind with the dragging effect of wagon trains and was hardened into a prejudice.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Then I spoke at the ninetieth birthday party of W. E. Burghardt Du Bois, who embarked on a fictional trilogy at eighty-nine and who, with The Crisis, had created a Negro intelligentsia that had never existed in America before him.

Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.

Then a full-time planning office will be established in Rome to work with a five-member Georgia Tech research staff for development of an area planning and industrial development program.

Then the Communese reply came back from many mouthpieces with striking consistency.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then Angelina turned and with an easy grace walked toward the kitchen.

Then she fell asleep again as soddenly as a person with fever, and when she awoke it was dark outside and the clarity was back in her eyes.

Then, abandoning the studies in the face of their promising outlook for all concerned, B. & O. entered on-again-off-again negotiations with C. & O. which resulted in the present situation.