Can radioactivity be slowed through time dilation? Can radioactivity be slowed using the effect of time dilation?
If you put cesium, tritium or uranium in a cyclotron at relativisitic speeds, do their half lives become longer in our frame?
Could this be used as a means to store radioactive material?
 A: 
Could this be used as a means to store radioactive material?

The volume or mass of material which could be stored this way would be extremely small. OTOH, there are radioactive ion beams used in experiments which might benefit from relativistic speeds in the beam line.
A literature search on relativistic radioactive ion beams reveals several experiments like this at relativistic energies, but the time dilation effects don't appear to be the primary motivation behind them, but rather the large energies which give better statistics for low cross-section reactions. See this paper.
They are also used specifically to have high-precision tests of special relativity: See this article.
But storage of large quantities of short-lived nuclides doesn't seem realistic.  The energy demands to keep them moving would be overwhelming and not cost effective. It's better to make them as you need them.
A: Yes. The classic example is that this is the only reason muons produced by cosmic radiation high up in the atmosphere live long enough to reach the ground.
A: The answer is yes but the amount of energy needed to generate a measurable time dilatation effect would be prohibitive. Let us say you put the material in the fastest centrifuge available today. The time dilatation would be on the order of a billionth of a second or less.
A: There's a simple mathematical argument as to why this might not be as useful as you'd think, even given a really cheap means of acceleration.
Let's say we have a relativistic "particle". It could be a fundamental particle or a space ship or a planet, doesn't matter. The time which passes in the lab frame is given by
$$
T = \gamma T_{0}
$$
where $T_{0}$ is the time that's passed in the particle frame, and $\gamma$ is
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
So if you have a cheap accelerator you can just make $v$ huge, and make $\gamma$ like 10,000. Now your radioactive thing lives 10,000 times longer, right?
Well, maybe that's not worth your time. The energy of your particle is given by
$$
E = \gamma m c^2 = \underbrace{(\gamma - 1) m c^2}_{\rm kinetic} + \underbrace{m c^2}_{\rm rest}
$$
where I broke the terms apart to show the rest energy and the kenetic energy. So for $\gamma \gg 1$, the kinetic energy is much larger than the rest energy.
The important part is that you can create your particle from thin air if you can produce the rest energy*. So for the energy that you used to extend the lifetime by a factor of 10,000, you could have created 9,999 of the same particle (by smashing some particles into a target with an energy equal to $m c^2$, for example).
This isn't to say that this would never be worth it: maybe the process to create your particle is really inefficient, and accelerating it is really cheap. But in general you're battling against the same factor of $\gamma$ that you're using to dilate time.

*Creating particles from thin air is subject to some conservation laws: you might have to create some byproducts in the process.
A: Yes, like the elementary particle travelling with speed of light gets extra life i.e. they last for longer period of time. In same way if somehow we can move radioactive substance faster than we will be able to expand the period of radioactivity. 
We know the relation,
T=0.693/$\lambda$ ,where T is half life and $\lambda$ is the decay constant.
 So as we move faster through space literally we move slower in time resulting change in value of $\lambda$. Thus we can change the value of $\lambda$ through time dilation.
