Radium, being radioactive, will irradiate and activate some things placed near a sample. The element radium in its "natural" form is an alpha particle emitter, and things that get hit by an alpha particle have a chance of undergoing nuclear transformation. An alpha particle, which is emitted by a 226Ra atom when it decays, is a helium-4 nucleus. It's composed of two protons and two neutrons. This is a "heavy hitter" as regards particulate radiation. It won't travel far, even in air, because it is too massive and it "runs into stuff" in scattering reactions because of its size. But when it reacts with a nucleus, things happen. That's how some materials near a radium source become radioactive.
The decay of radioactive substances follows a decay chain that will sooner or later result in the appearance of a stable isotope of lead. There is an exception for the atoms of a few substances that have undergone decay by spontaneous fission.
Uranium-238, or more properly 92238U, is naturally radioactive. (It does not "become" radioactive.) Radioactivity of an isotope simply means that it has an unstable nucleus. It is unstable because the nucleus is large enough that the nuclear force (residual strong atomic force) that holds the nucleus together is offset by the competing electromagnetic force which makes protons repel each other. The reason for this offset is that the nuclear force declines with distance at a greater rate than the electromagnetic force. There are other reasons for radioactivity, such as isotopic variations in neutron to proton ratio, but the size of the nucleus is primary, for elements with atomic number greater than 82.
If you are referring to a cell's nucleus than the simple answer is that's not radioactive. Radioactivity occurs when elemental atoms become unstable due to the loss or gain of additional neutrons; these unstable atoms are referred to as radioactive isotopes. If a cell's nucleus were radioactive it would not last very long, its structure and function would quickly degrade and collapse.
The process described is radioactive decay.
it become zero
This phenomenon is rare, non obligatory. For example beryllium irradiated with alpha particles from radium emit neutrons.
Radioactive substances have half-lives. This is because the isotope constantly is changing from the radioactive isotope to a daughter element. For example, eventually, when uranium's radioactivity is gone, it becomes lead. After one half life of a radioactive substance, only 50% of that substance is still radioactive. Therefore, after one half-life, a piece of uranium is 50% lead and therefore %50 less radioactive. After another half-life, it has 25% of the original radioactivity, and 75% of the original uranium has become lead. This is the problem with radioactive wastes. It takes many years just for one half lives for some substances, such as uranium. Because radioactivity is harmful, those substances have to be stored until they are no longer radioactive. So, in short, the problem with disposing of radioactive wastes is that they have long half-lives. (although this is not true with ALL substances because some have short half-lives, but, in general, radioactive substances have long half-lives.
We can make nonradioactive substances radioactive by exposing them to particulate radiation of some kind. (Electromagnetic radiation like gamma rays won't work.) There are sever kinds of particulate radiation, and they include neutrons, protons and alpha particles as well as beta particles. Exposure of a material to these particles allows the particles to activate the material, and the result of activation will depend on the radiation being used and the reaction of the substance to that activating particle flux. Just to supply one example, if we take a slug of cobalt (cobalt-59) and lower it into a working nuclear reactor, the neutron flux will bathe the cobalt. Neutrons will be absorbed by some of the cobalt atoms and they will transform into cobalt-60, which is an unstable isotope of cobalt. It's radioactive, and is a gamma ray emitter. We then withdraw the slug (now called a source) and put it in a "vault" to store it, and we use the slug (opening the vault door by remote control) as a radiation source to, say, perform X-ray analysis of pipe welds in the field where dragging an X-ray machine to the job isn't practical.
Not always -- Hydrogen-3 is radioactive, for example.
This region become a radioactive contaminated area.
Patients receiving brachytherapy do become temporarily radioactive
By becoming unstable
The decay of radioactive substances follows a decay chain that will sooner or later result in the appearance of a stable isotope of lead. There is an exception for the atoms of a few substances that have undergone decay by spontaneous fission.
Patients receiving intracavitary radiation do become temporarily radioactive
Patients receiving implant therapy do become temporarily radioactive
Patients receiving interstitial radiation do become temporarily radioactive
The bacterial proteins will become radioactive